Quantifying Extinctions

University of South Wales

Final Year Project

Quantifying Extinctions

Author:

Benjamin Rowe

Supervisor:

Dr. Graeme Boswell

April 30, 2015

Abstract

The Red List is a database of all the different species what have been evaluated in terms

of their risk of extinction, the aim of this report is to quantify the risk of extinction to help

supplement the data currently used in the Red List. This project, in particular, focuses

on times to extinction and population distributions using a combination of analytical

mathematics and computer simulations. The Results were also backed up with random

simulation. The results showed that population distributions are not affected by catas-

trophes but times to extinction were heavily reduced with the inclusion of catastrophes,

also the introduction of random error made significant differences. This report suggests

that it is essential to consider catastrophes when classifying the risk of extinction of a

species.

UNIVERSITY OF SOUTH WALES

PRIFYSGOL DE CYMRU

FACULTY OF COMPUTING, ENGINEERING & SCIENCE

SCHOOL OF COMPUTING & MATHEMATICS

STATEMENT OF ORIGINALITY

This is to certify that, except where specific reference is made, the work described in

this project is the result of the investigation carried out by the student, and that neither

this project nor any part of it has been presented, or is currently being submitted in

candidature for any award other than in part for the degree of BSc or BSc with Honours

of the University of South Wales.

Signed .................................................................................... (student)

Date ....................................................................................

(This statement must be bound into each copy of your Project Report.)

1

Acknowledgments

The one person I would like to thank for not just being my project leader, but also

being my tutor in first year and having taught me all three years throughout University

is Dr. Graeme Boswell. He has helped me to reach my potential each year while at the

same time making me work for it and for that I am beyond grateful.

2

Contents

1 Introduction 6

1.1 How Species are Classified [5] . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Least Concern (LC) . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.2 Near Threatened (NT) . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 Vulnerable (VU) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.4 Endangered (EN) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Critically Endangered (CR) . . . . . . . . . . . . . . . . . . . . . 10

1.1.6 Extinct in the Wild (EW) . . . . . . . . . . . . . . . . . . . . . . 10

1.1.7 Extinct (EX) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Discrete Time Models 12

2.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 The Initial Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Expected Times to Absorption . . . . . . . . . . . . . . . . . . . . 16

2.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Birth and Death Rates . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Population Distributions without Catastrophes . . . . . . . . . . . . . . . 19

2.3.1 Expected Times to Extinction . . . . . . . . . . . . . . . . . . . . 20

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Continuous Time Models: Times to Extinction 26

3

3.1 Expected Time to Extinction . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Population Distributions without Catastrophes . . . . . . . . . . . . . . . 29

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Continuous Time Models: Catastrophes 31

4.1 Expected Time to Extinction with Catastrophes - Derivation . . . . . . . 31

4.2 Expected Time to Extinction with Catastrophes - Results . . . . . . . . . 33

4.3 Expected Population Distributions . . . . . . . . . . . . . . . . . . . . . 35

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Simulation with NetLogo 42

5.1 Code Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 Expected Times to Extinction . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Population Distribution . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Brownian Motion 48

6.1 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2 Sample Path Equation Derivation . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Conclusions 57

A NetLogo Code 63

A.1 Initial Simulation.nlogo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B MatLab Code 68

B.1 Dissertation.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.2 Binomial.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4

B.3 Continuum Matrix.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C Maple Code 73

C.1 V Graphs.mw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5

Chapter 1

Introduction

There have been many extinctions in the world, some more commonly known like the

Dinosaurs or the Dodo, and some less known. A species is not officially classed as extinct

until it has not been seen in the wild for at least 50 years. There are isolated extinctions

(e.g. extinct in Britain but still live in Europe) or mass extinction (very rare), a species

can also be extinct from the wild if it only exists in captivity. A functionally extinct

species is when there are no members of that species population that are capable of re-

producing.

The ’Heath Hen on Martha’s Vineyard’ is a good example, living on the coasts of North

America. It was noted around 1830 that their numbers were starting to rapidly decline

and by 1870 their population was believed to be restricted to Martha’s Vineyard. In the

early 20th century there was only about 50 hens left and in 1908 a sanctuary was created

to try and increase population numbers. In the next 5 years or so the 50 in the beginning

rose to over 2000 but a poultry disease brought from domestic turkeys and only 13 living

heath hens were recorded to be living in 1927. The population never recovered after this

and a couple of natural disasters, the last heath hen, Booming Ben, died in 1932. [4]

The story displays examples of human influence, predators, natural disasters, isolated

extinction and functional extinction.

6

There are many causes for extinction, like starvation, natural disasters/catastrophes,

over hunting, predators or competition, all of which can be objectively modelled mathe-

matically to classify the status of the species, the only problem is coming up with different

definitions on values and expected times to extinction which in itself can be subjective.

Quantifying the risk of these extinctions is important because it helps the human race

to understand a suitable point in time where they should start intervening (i.e. breeding

programs or relocation or to stop the hunting). Without these barriers there would be

many more animals already not in existence, on the other hand sometimes if a popula-

tion is too big it can indirectly affect other species, hence a cull would also come into

action. There is a set of classifications already in place to distinguish how ’well’ a species

is surviving Figure (1.1).

Figure 1.1: Icons to classify status of species population, also known as the red list [1].

These letters stand for (from right to left): Extinct; Extinct in the Wild; Critically En-

dangered; Endangered; Vulnerable; Near Threatened, Least concern; there are two more

classifications where the species at hand hasn’t been evaluated or the data available isn’t

sufficient. [3]

1.1 How Species are Classified [5]

The Red List is a set of criteria that evaluate any species on population numbers if there

is a believed risk of numbers making a significant drop, after assessing the required data

against the criteria the species will then be put into one of the classes. ’Least Concern’

and ’Near Threatened’ there will be almost never any action taken, the next 3 classes

7

action is taken in order to raise numbers with the hope of putting them back into one

of the first two classes. The last two ’Extinct in the Wild’ and ’Extinct’ are extreme

concerns but luckily quite rare.

1.1.1 Least Concern (LC)

This is the lowest level of conservation, species that fall in this category (e.g. humans or

houseflies) have a very widespread population.

1.1.2 Near Threatened (NT)

A near threatened species are likely to fall under a threatened status in the near future.

1.1.3 Vulnerable (VU)

The next few categories are based on 5 different criteria: population reduction rate;

geographic range; population size; population restrictions; and probability of extinction.

An example of a vulnerable species is the ’Snaggletooth Shark’, while its area of occupancy

is massive the rate of reduction in their population numbers is more than 10% in the last

10 years.

To be classed as vulnerable, the population of that species must meet any of the criteria

below:

1. The population has declined between 30-50 percent in the last 10 years or 3 gener-

ations (whichever is longer)

(a) 30% if the cause of the decline is unknown.

(b) 50% if the cause of the population decline is known.

2. The population’s:

(a) ’extent of occurrence’ is less than 20, 000 square kilometres; the extent of

occurrence is the minimum area in which it could contain all the different sites

to which that species lives on.

8

(b) ’area of occurrence’ is less than 2000 square kilometres.

3. If the species has less than 10,000 mature individuals present.

4. If the restriction on the species is less than 1000 mature individuals per 20 square

kilometres.

5. If the probability of extinction is at least 10% in the next 100 years.

1.1.4 Endangered (EN)

An example of an endangered population is the ’Siberian Sturgeon’. Overfishing and

poaching has meant that numbers have declined between 50 and 80 percent in the last

three generations (about 50 years).

A species is classed as endangered if any of the following points are met:

1. The population has been reduced between 50 and 70 percent over the last 10 years

or three generations.

(a) 50% if the cause of population reduction is unknown.

(b) 70% if the cause of loss of population in known.

2. The species has a:

(a) ’Extent of occurrence’ of less than 5,000 kilometres squared.

(b) ’Area of occupancy’ of less than 500 kilometres squared.

3. There are fewer than 2,500 mature individuals. Also the species is classed as en-

dangered if the population has declined more than 20% in the last 5 years (or 2

generations).

4. When the population in one area of occupancy is less than 250 mature individuals,

at this point the extent of occurrence is not considered.

5. If the probability of extinction is more than 20% in the next 20 years or 5 generations

(whichever is longer).

9

1.1.5 Critically Endangered (CR)

An example of a critically endangered species is the Bolivian Chinchilla Rat, this is due

to the fact that its extent of occurrence is even less than 100 square kilometres. The

forests these rodents live in are being cut down to make room for livestock.

To be classed as critically endangered a species has to meet one of the following criteria:

1. Measured over whichever is longer out of 10 years and three generations, if the

population has declined in numbers of at least 90% or 80% if the cause of the

decline is still unknown.

2. If the population’s:

(a) extent of occurrence is less than 100 kilometres squared.

(b) area of occupancy is less than 10 square kilometres.

3. If there are less than 250 mature individuals left.

4. When each of the populations is restricted to just 50 mature individuals.

5. When the probability of extinction is at least 50% in the next 10 years.

1.1.6 Extinct in the Wild (EW)

A species that is extinct in the wild, only exists in cultivation (plants) and captivity

(animals) so for example a closely monitored garden or a zoo. However, a species is only

recorded as EW if there have been no studies to confirm numbers in the wild after a

number of years.

One example of a species that is EW is the Black soft-shell turtle, a small population of

150-300 live in a pond near a place of worship and hence are not allowed to be taken, this

has resulted in no programs to help the populations in the wild or many being found in

zoos around the world.

10

1.1.7 Extinct (EX)

While it is very hard to prove a species is definitely extinct, a species is classified as

extinct when there are no more left in the wild or captivity, normally declared after nu-

merous studies and no reported sightings after an extensive period of time.

The problem at hand is that the criteria ’Probability of Extinction’ is just a percent-

age, and again is based upon a deterministic approach. The aim of this project is to see

if there is a suitable replacement for this criteria that not only gives a time the aver-

age person can understand but also introduces an element of randomness (or noise) that

eventually leads to introducing confidence limits. With all this it would then be possible

to give a level of confidence depending on the set constraints and alter the confidence in-

tervals as needed (e.g. 99% confidence limits will be more widespread about the estimate

than 60% for example).

In this report the problem at hand will be tackled in gradual steps. First the birth

and death rates alone will be considered with a discrete time variable. After Discrete

time has been evaluated a continuum approach will then be investigated where eventu-

ally catastrophe rates will also be included. After these main rates have been included

a stochastic approach is considered as well as a some computer simulations to hopefully

give more evidence from a numerical sense.

11

Chapter 2

Discrete Time Models

Discrete data, while not the most appropriate method, will give data easier to work with

and can be used to make future predictions. Also once the basics are known from the

analysis of this data then applications can be made later on in future chapters.

2.1 Illustrative Example

2.1.1 The Initial Problem

Consider a counter performing an unbiased random walk on a simple number line (Figure

2.1) starting from 0 and going up to 4 is moved. A ’fair’ coin is flipped every minute;

if the coin lands head side up then the counter is moved one position to the right and

to the left if the coin lands tails side up. If a heads is flipped while the counter is on

position 4 then the counter does not move position. The game ends when the counter

reaches position 0.

There are two questions at hand: (1) what is the probability of the counter being at

a particular position at some time t?; and (2) what is the expected time to position 0

from each unique starting point of the counter on the number line?

The fractions labelling the arrows represent the probabilities of moving from each posi-

12

Figure 2.1: An unbiased random walk on {0, 1, 2, 3, 4}.

tion to the next at the end of each minute (Figure 2.1).

2.1.2 Probability Distributions

Suppose, for example, the counter starts off at position 1 at time t = 0. After n minutes

(i.e. n jumps), the position of the counter can be described by a probability distribution

determined from the incidence matrix of the random walk.

P =

1 0.5 0 0 0

0.5 0 0.5 0 0

0 0.5 0 0.5 0

0 0 0.5 0 0.5

0 0 0 0.5 0.5

(2.1)

Note that Pij denotes the probability of moving from position ’j − 1’ to position ’i − 1’

at the end of each minute.

The following vectors Vn represent the probability distribution of the counter being in

that position after n minutes. V0 represents the counter being placed at position 1 to

start the game off.

13

V0 =

0

1

0

0

0

(2.2)

V1 = PV0 =

0.5

0

0.5

0

0

V2 = PV1 = P 2V0 =

0.5

0.25

0

0.25

0

V3 = PV2 = P 3V0 =

0.625

0

0.25

0

0.125

V4 = PV3 = P 4V0 =

0.625

0.125

0

0.1875

0

(2.3)

14

limn→∞

Vn = P nV0 =

1

0

0

0

0

(2.4)

This demonstrates that 0 is absorbing, even starting at different positions doesn’t change

the final result; i.e. the initial data doesn’t change the final outcome. This can be proven

with the power method of matrices.

Taking the eigenvalues using MatLab and arranging them into a diagonal matrix D:

P = C−1DC (2.5)

1 0.5 0 0 0

0 0 0.5 0 0

0 0.5 0 0.5 0

0 0 0.5 0 0.5

0 0 0 0.5 0.5

= C−1

1 0 0 0 0

0 −0.766 0 0 0

0 0 −0.1736 0 0

0 0 0 0.5 0

0 0 0 0 0.9329

C (2.6)

and irrespective of what C is

P k = C−1

1 0 0 0 0

0 −0.766 0 0 0

0 0 −0.1736 0 0

0 0 0 0.5 0

0 0 0 0 0.9329

k

C (2.7)

So

limk→∞

P k = limk→∞

C−1DkC (2.8)

15

=

1 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

(2.9)

And so because only that one value remains for high values of k this demonstrates that

state 0 is an absorbing state, this could have also been demonstrated from:

λ =

1

0

0

0

0

(2.10)

2.1.3 Expected Times to Absorption

In the previous section it was seen that 0 was an absorbing state; irrespective of the

starting position, the counter would always end up at 0. The only difference made by the

starting state is the time taken to reach 0. Using an approach similar to the above, it is

possible to calculate this expected time.

Let E[J ] denote the expected time for the random walk starting from position J to

reach position 0. Consider the counter starting at position 1. After 1 minute with prob-

ability a half the counter moves to the right and then the expected time changes to the

expected time from position 2 but with probability 12

the counter can go left to position

0 and at which point the random walk has reached the absorbing state. I.e.:

E[1] = 1 +1

2E[2] +

1

2E[0] (2.11)

and since E[0] = 0

16

E[1] = 1 +1

2E[2]

Other states are defined similarly:

E[2] = 1 +1

2E[1] +

1

2E[3]

E[3] = 1 +1

2E[2] +

1

2E[4]

E[4] = 1 +1

2E[3] +

1

2E[4]

Rearranging these:

E[1]− 1

2E[2] = 1

−1

2E[1] + E[2]− 1

2E[3] = 1

−1

2E[2] + E[3]− 1

2E[4] = 1

−1

2E[3] +

1

2E[4] = 1 (2.12)

which can easily be represented in matrix form as

1 −12

0 0

−12

1 −12

0

0 −12

1 −12

0 0 −12

12

E[1]

E[2]

E[3]

E[4]

=

1

1

1

1

(2.13)

Notice the expected times of Equation 2.6 are of the form: BE = C where B and C are

known. Then E an be obtained by multiplying both sides by B−1 to get:

E = B−1C (2.14)

Using MatLab:

E = B−1C =

2 2 2 2

2 4 4 4

2 4 6 6

2 4 6 8

1

1

1

1

=

8

14

18

20

(2.15)

17

So the expected times to reach state zero starting from the different states are:

E[1] = 8 minutes

E[2] = 14 minutes

E[3] = 18 minutes

E[4] = 20 minutes (2.16)

2.2 Application

If we were to assign the position number in the above illustrative example to the number

of females in a population at time t (which could be in days, weeks, or even years) then

moving to the left corresponds to one female dying and moving to the right corresponds

to the birth of another female. This is the most extreme simple case possible as it only

allows for one to die or be born each time step which is very unrealistic, also that the

probabilities are equal in that they are both one half.

To further increase the accuracy multiple births and multiple deaths alongside each other

could be considered (although it is unnecessary). We will first consider the population

details.

2.2.1 Birth and Death Rates

In this model we are going to assume that the probability of moving right from position

j is Rj and moving to the left is Qj, so Rj is the probability that there is a birth when

there are j members of the population, and Qj is the probability of a death when there

are j members. Suppose

Rj = bj

Qj = dj2 (2.17)

where b and d are small and which is in keeping with the logistic model of population

growth. Further suppose there is a maximum population (K) where RK = 0.

18

2.3 Population Distributions without Catastrophes

Consider a Markov chain representing a population where the states correspond to the

number of individuals. As the population’s extinct in state zero it is not included. This

Markov chain is represented in Figure 2.2 and Figure 2.3. ’b(2)’ represents the probability

of a birth from state 2 and ’d(2)’ therefore represents the probability of a death from state

2.

Looking at these and assuming that the system is in balance (i.e. the number leaving

Figure 2.2: Markov Chains representing system of births and deaths where K = 3.

Figure 2.3: Markov Chains representing system of births and deaths where K = 4.

a state is matched by what’s entering it), it is then possible to use these probabilities to

estimate the probability distribution of the population.

About state 1:

P (1)b(1) = P (2)d(2)

P (1) =P (2)d(2)

b(1)(2.18)

About state 2:

P (1)b(1) + P (3)d(3) = P (2)[d(2) + b(2)]

19

P (2)d(2) + P (3)d(3) = P (2)[d(2) + b(2)]

P (3)d(3) = P (2)b(2)

P (2) =P (3)d(3)

b(3)(2.19)

About state 3:

P (2)b(2) + P (4)d(4) = P (3)[b(3) + d(3)]

P (3)d(3) + P (4)d(4) = P (3)[b(3) + d(3)]

P (4)d(4) = P (3)b(3)

P (3) =P (4)d(4)

b(3)(2.20)

From here a pattern has started emerging, and so in a general form:

P (a) =P (a+ 1)d(a+ 1)

b(a)∀1 ≤ a ≤ K − 1 (2.21)

The first step of using this equation is to estimate a value for P(K) and apply Equation

(2.21) iteratively to calculate P(1)...P(K). Then if:

K∑i=1

P (i) 6= 1 (2.22)

A second iteration is needed but this time set

P (K) =estimated P (K)∑K

i=1 P (i)(2.23)

to ensure the sum of probabilities of being at each state all add up to equal 1, this makes

sure P(i) is now a genuine probability distribution.

This approach can be very time consuming, especially with larger values of K, so a

MatLab code was produced in order to do this efficiently and quickly (see Appendix B.2.

This created Figures 2.4, 2.5 and 2.6.

2.3.1 Expected Times to Extinction

Given the illustrative example seen when calculating the expected times to absorption,

it is clear that the population will become extinct eventually. However, it is possible to

20

Figure 2.4: Distribution Graphs found by solving Equation (2.21) with K = 10.


determine the probability distribution of the population before its extinction via a simple

argument based on movements between states.

Like before, assuming that only one event can occur in any one time step but now

using more realistic probabilities for births and deaths we end up with a similar matrix

21


formation as before in Equation (2.13).

1 −bj 0 · · · 0

−dj2 1 −bj · · · 0

· · · −dj2 1 −bj · · ·

0 0 −dj2 1 · · ·

E[1]

E[2]

E[3]

· · ·

E[j = n]

=

1

1

1

· · ·

1

(2.24)

This work is based on a paper by B. J. Cairns and P. K. Pollett [6] where matrix manip-

ulation is used to calculate expectancy rates.

Where Equation (2.24) is a symmetrical matrix. From here this can be rearranged and

solved for the expected times to extinction, for discrete data, can be calculated in the

form of...

E[1]

E[2]

E[3]

· · ·

E[j = n]

=

1 −bj 0 · · · 0

−dj2 1 −bj · · · 0

· · · −dj2 1 −bj · · ·

0 0 −dj2 1 · · ·

−1

1

1

1

· · ·

1

(2.25)

22

With Figures 2.7, 2.8 and 2.9, these display what would be the expected times to extinc-

Figure 2.7: Times to Extinction from state j found by using Equation (2.25).


tion in unit time when the target population is currently in state j. The main problem

straight away is that it is oscillating around the zero (or extinct) level, in practice it

would be assumed that once it reaches 0 it will stay at zero and not just simply go from

being extinct to being alive again.

23


In the cases previously looked at K has been the cap on the population (or more applied

could be a fixed carrying capacity), from this and looking at the distributions it is possi-

ble to find the state that the population is most likely to be in (Figures 2.4..2.6 suggests

all three to be around 5 or 6) and then use the expected time for that state from Figures

2.7, 2.8 and 2.9 to estimate a time until extinction (again, in unit time). So some of the

results from the graphs in Figures 2.7..2.9 are below:

E[K] = −20.2348 (2.26)

E[µ] = 9.4228 When K = 10 (2.27)

E[K] = 903.6297 (2.28)

E[µ] = −49.9629 When K = 15 (2.29)

E[K] = 886.8743 (2.30)

E[µ] = 0.3738 When K = 20 (2.31)

24

In the results above, µ is the mean number in the population with an upper boundary

(cap) of K. This mean number was drawn from Figure 2.4.

These values where results from using Equation (2.25) with b = 0.1, and d = 0.02,

and are also measured in unit time.

2.4 Discussion

The first problem with the above is that for values of K under 20 it fails to produce

a realistic result, in that they are all negative and this makes no sense when it comes

to approximating times to extinction. These results are most likely due to the fact that

a discrete time step has been used and isn’t an accurate enough measure for what we need.

Another problem may arise with respect to the fact that a discrete set of data was

used against distribution graphs that are based on a continuum approach.

25

Chapter 3

Continuous Time Models: Times to

Extinction

In Chapter 2 time was modelled in discrete steps/chunks, the problem at hand is what

if multiple events could happen in each time step and hence a continuum approach is

required.

3.1 Expected Time to Extinction

Suppose time (t) is now continuous, where ∆t is sufficiently small, and there can be a

number of different events including: a birth; a death; or nothing happens at any point.

At this stage with a current population ’j’, let E[j] denote the expected time to extinc-

tion from j, b(k) and d(j) are also the birth and death rates.

So it can be modelled that the expected time to extinction is after a period of time

either a birth happens and the expected time to extinction is now E[j + 1], a death

happens and the expected time to extinction is now E[j−1], or nothing happens and the

expected time to extinction stays the same. In mathematical terms it can be represented

by:

26

E[j] = ∆t+ b(j)∆tE[j + 1] + d(j)∆tE[j − 1] + (1− b(j)− d(j))∆tE[j] (3.1)

0 = ∆t+ b(j)∆tE[j + 1] + d(j)∆tE[j − 1]− (b(j) + d(j))∆tE[j] (3.2)

Then divide by ∆t and rearrange

−1 = b(j)E[j + 1] + d(j)E[j − 1]− (b(j) + d(j))E[j] (3.3)

Putting this last equation into matrix form makes it possible to solve in a similar manner

to the previous chapter for E. Then putting a ’cap’ on the system (k) so that b(K) = 0

ensures there is a finite number of equations when E[0] = 0.

k∑j=0

= 1 (3.4)

3.2 Numerical Solutions

Putting Equation (3.3) into matrix form and solving for all E[j] with 1 ≤ j ≤ K gives

you Figures 3.1, 3.2 and 3.3. Using the data from Figures 3.1, 3.2 and 3.3 and their

corresponding distribution graphs in Figures 2.4, 2.5 and 2.6 some values can be found

for the expected time to extinction with a carrying capacity of K.

Using a similar approach as in the previous chapter, in matrix form it can be repre-

sented as:

E[1]

E[2]

E[3]

· · ·

E[j = K]

=

−bj − dj2 bj 0 · · · 0

dj2 −bj − dj2 bj · · · 0

· · · dj2 −bj − dj2 bj · · ·

0 0 dj2 −bj − dj2 · · ·

−1

−1

−1

−1

· · ·

−1

(3.5)

Looking at Figures 2.4, 2.5 and 2.6, the value of µ (where µ is the average number of

mature individuals in the population) was found by looking for the highest value in the

27

graphs and then comparing this value of µ with Figures 3.1, 3.2 and 3.3 then respectively

to get the expected times to extinction.

E[K] = 487.3401

E[µ] = 479.5891 when K = 10

E[K] = 492.6602

E[µ] = 481.9846 when K = 15

E[K] = 493.8111

E[µ] = 481.9932 when K = 20 (3.6)

Looking at these values it is noticeable that the numbers increase slower and slower, this

is unlike what happened in Chapter 2 where as K increased the values of the graph shot

up to very large values very quickly. Figures 3.1, 3.2, and 3.3 were solved using B.1.

Figure 3.1: Graphs showing the expected times to extinction found by solving Equations

(3.3) and (3.4) with b = 0.1, d = 0.02 and K = 10.

28


(3.3) and (3.4) with b = 0.1, d = 0.02 and K = 15.


(3.3) and (3.4) with b = 0.1, d = 0.02 and K = 20.

3.3 Population Distributions without Catastrophes

Assuming if the value of time ∆t was made small enough then the Markov chains used

in Section 2.3 would be exactly the same and hence the same equations and distribution

29

graphs would be this result. The same graphs are produced also because there are no

catastrophes included at the moment meaning no extra variables have been included.

3.4 Discussion

So using continuous data and assuming ∆t to be small enough results in a completely

different expected times graph compared to when discrete data was used, although the

population distribution remains unchanged. With the work done in this chapter, a new

set of expected times (E[µ]) in Equation 3.6.

Now compared to Chapter 2, both the expected times graphs and distribution graphs

are both based on continuous data and the results all make a lot more sense. In Figures

3.1, 3.2, and 3.3 all the times to extinction are now positive and the curves in the graphs

make a lot more sense. The shape of these graphs are more realistic because when the

numbers of mature individuals starts to diminish the expected times drop significantly,

and as the numbers start to thrive the times increase slower and slower (i.e. as population

numbers → K, the gradient of the graph → 0).

Remembering back with the example of the Heath Hen of Martha’s Vineyard in Chapter

1, there were many examples of natural disasters (or even man-made disasters) which

will be referred to in this report as catastrophes. The next Chapter looks into these

catastrophe rates alongside the continuum approach and on top of the birth and death

rates. After finding new solutions we can compare the population distributions and ex-

pected times to extinction where catastrophes weren’t present. Catastrophes are relevant

in this work mainly because so far during each small time period only one death/birth

has occurred, this will now look into multiple occasions in each unit time.

30

Chapter 4

Continuous Time Models:

Catastrophes

In Chapter 3 while a continuum approach was considered, only a maximum of one death/

birth in each unit time was permitted. In chapter 4 catastrophes are going to be consid-

ered alongside several deaths at the same time too, however for simplicity at the moment

only a maximum of one birth is allowed in each unit time.

We are looking mainly into catastrophes to add an element of randomness to the es-

timated time to extinction that was discussed in the introduction, at the moment the

probability to extinction is just a percentage so the aim here is to create a more accu-

rate estimate and maybe add some confidence limits (or noise) to what is believed to be

correct.

4.1 Expected Time to Extinction with Catastrophes

- Derivation

In Chapter 3 a system of equations were created (Equation (3.3)) assuming there were a

single birth, a single death or no population change over a small time interval. Looking

31

back to Equation (3.1) we had:

E[j] = ∆t+ b(j)∆tE[j + 1] + d(j)∆tE[j − 1] + (1− b(j)∆t− d(j)∆t)E[j] (4.1)

From here to change it from a deterministic approach to a stochastic one we need to add

an element of catastrophe, hence:

E[j] = ∆t + b(j)∆tE[j + 1] + d(j)∆tE[j − 1] + (1− b(j)∆t− d(j)∆t)E[j]

+ ∆ta∑i=1

(K∑

j=a+1

P (j)q(i|j))b(a) (4.2)

Next Equation (4.2) needs to be rearranged to be able to work with it.

0 = ∆t + b(j)∆tE[j + 1] + d(j)∆tE[j − 1]− (b(j)∆t+ d(j)∆t)E[j]

+ ∆ta∑i=1

(K∑

j=a+1

P (j)q(i|j))b(a)

0 = 1 + b(j)E[j + 1] + d(j)E[j − 1]− (b(j) + d(j))E[j] +a∑i=1

(K∑

j=a+1

P (j)q(i|j))b(a)

−1 = b(j)E[j + 1] + d(j)E[j − 1]− (b(j) + d(j))E[j] +a∑i=1

(K∑

j=a+1

P (j)q(i|j))b(a)

(4.3)

Which can now be arranged into matrix form where,

M = BE

E = B−1M (4.4)

Where...

E =

E[1]

E[2]

E[3]

· · ·

E[K]

, M =

−1

−1

−1

· · ·

−1

(4.5)

B =

−b(1)− d(1)2 b(1) · · · 0

d(1)2 + q(1|2) −b(2)− d(2)2 b(2) · · ·

· · · d(2)2 + q(2|3) −b(3)− d(3)2 b(3)

q(i|K) · · · d(K − 1)2 + q(K − 1|K) −b(K)− d(K)2

(4.6)

32

At this point it should be explained that q(i|j) is the catastrophe term included in the

matrix, where q is the probability that a catastrophe occurs and the (i—j) is related back

to the binomial distribution where:

q(i|j) = q

j

i

pj−i(1− p)i (4.7)

j is the number of individuals present when the catastrophe occurs, i is the number of

individuals left after the catastrophe has happened and p is the probability that each

individual dies when the catastrophe happens.

In Equation (4.6) ’i’ corresponds to the column number and ’j’ holds the same value

as the row number of the position it holds in the matrix. Anything below, and not in-

cluding, the subdiagonal is just the catastrophe term and does not hold any death terms.

All values above, and not including, the superdiagonal are just equal to zero.

4.2 Expected Time to Extinction with Catastrophes

- Results

The main difference in the equations considering catastrophes compared to the case with

no catastrophes (Chapter 3) was the inclusion of the entries in the matrix below the

diagonal. Solved using MatLab and the code in Appendix B.1 the q term can be set to

zero to eliminate any catastrophes. The presence of these terms can cause a significant

difference in the expected times to extinction from each state, as seen in Figures 4.1, 4.2

and 4.3.

With more of a chance of dying due to catastrophes this would make a lot of sense

that the times have reduced considerably. Comparing some of the values between with

catastrophes and without catastrophes there is a difference of around 120 between equiv-

alent states, also considering that the probability of a catastrophe occurring is only one

in a thousand this could drastically decrease expected times to extinction further if this

33

Table 4.1: Table comparing catastrophe (q) and catastrophe-death (p) rates with the

average expected time to extinction over 20 simulations

p 0.001 0.5 0.001 0.1 0

q 0.2 0.2 1 0.7 -

Av. 152.667 15.333 90.853 27.417 149.5

value was even slightly increased.

In Table 4.2 there is 5 different combinations of values for p and q and below is the aver-

age time to extinction when these values were put into NetLogo (a stochastic simulation

program). The average can be assumed to be a number of small time intervals, more will

be discussed on NetLogo in the next chapter. Figures 4.1, 4.2 and 4.3 were plotted using

Appendix B.3 in MatLab.

Figure 4.1: Graphs showing the expected times to extinction found by solving Equation

(4.6) with b = 0.1, d = 0.02, q = 0.001, and K = 10.

34


(4.6) with b = 0.1, d = 0.02, q = 0.001, and K = 15.


(4.6) with b = 0.1, d = 0.02, q = 0.001, and K = 20.

4.3 Expected Population Distributions

The expected times to extinction alone aren’t of much use, even though a carrying capac-

ity may have been set it is still unknown where the population’s average state is found

35

below this, if the birth rates are high enough then the population will be close to the

K value. If the population’s catastrophe and or death rates are high enough the system

may hang around the lower valued states.

Taking the same approach as last time by just trying to come up with a set of simulta-

neous equations proven itself very ’fiddly’ and difficult so another approach is required,

we first consider a population that can consist of at most 3 individuals (i.e. the Markov

chain has 3 states). This system is seen in Figure 4.4.

From here, seeing as the population is bounded by 3, we see that P(1) + P(2) +

Figure 4.4: Markov Chains representing system of births, deaths and catastrophes where

K = 3.


K = 4.

P(3) = 1 in a state of equilibrium where P(j) is the probability there are j individuals in

36


K = 5.

the population. Setting these probabilities equal to each other and re-arranging gives:

About state 1:

P (1)b(1) = P (2)d(2) + P (2)q(1|2) + P (3)q(1|3)

P (1) =P (2)[d(2) + q(1|2)] + P (3)q(1|3)

b(1)(4.8)

About state 2

P (2)[b(2) + d(2) + q(1|2)] = P (1)b(1) + P (3)[d(3) + q(2|3)]

= P (2)[d(2) + q(1|2)] + P (3)[d(3) + q(1|3) + q(2|3)]

P (2)[b(2) + d(2)− d(2) + q(1|2)− q(1|2)] = P (3)[d(3) + q(1|3) + q(2|3)]

P (2)[b(2)] = P (3)[d(3) + q(1|3) + q(2|3)]

P (2) =P (3)[d(3) + q(1|3) + q(2|3)]

b(2)(4.9)

About state 3

P (3)[d(3) + q(2|3) + q(1|3)] = P (2)b(2)

= P (3)[d(3) + q(1|3) + q(2|3)]

P (3)d(3) = P (3)d(3) (4.10)

From this result it shows that the value of P (3) determines the value of P (1) and P (2).

However as stated before,

n∑i=1

P (i) = 1 (4.11)

37

where n is the cap value of the system. With this data a value of P(3) can be chosen

to ensure that P (1) + P (2) + P (3) = 1. In practice an estimated value of P(3) is first

selected and if equation 4.4 does not equal 0 then,

P (n) =estimated P (n)∑n

i=1 P (i)(4.12)

After this second part of the algorithm then the values are final, however a check to make

sure Equation (4.11) is now satisfied is always preferred.

Now consider the case with a population comprising of at most 4 individuals (Figure

4.5).

About state 1:

P (1)b(1) = P (2)[d(2) + q(1|2)] + P (3)q(1|3) + P (4)q(1|4)

P (1) =P (2)d(2) +

∑4i=2 P (i)q(1|i)

b(1)(4.13)

About state 2

P (1)b(1) + P (3)[d(3) + q(2|3)] + P (4)q(2|4)

= P (2)[d(2) + q(1|2) + b(2)]

= P (2)[d(2) + q(1|2) + b(2)]

= P (2)[d(2) + q(1|2)] +4∑i=3

P (i)q(1|i)

+ P (3)d(3) + P (3)q(2|3) + P (4)q(2|4)

= P (3)d(3) +4∑i=3

P (i)q(1|i) +4∑i=3

P (i)q(2|i)

P (2) =P (3)d(3) +

∑4i=3 P (i)q(1|i) +

∑4i=3 P (i)q(2|i)

b(2)

=P (3)d(3) +

∑4i=3[P (i)q(1|i) + P (i)q(2|i)]

b(2)(4.14)

About state 3

P (2)b(2) + P (4)d(4) + P (4)q(3|4) = P (3)[d(3) + b(3) + q(1|3) + q(2|3)]

P (3)d(3) +4∑i=3

[P (i)q(1|i) + P (i)q(2|i)] + P (4)d(4) + P (4)q(3|4)

38

= P (3)d(3) + P (3)b(3) + P (3)q(1|3) + P (3)q(2|3)

P (3)b(3) = P (4)q(1|4) + P (4)q(2|4) + P (4)d(4) + P (4)q(3|4)

P (3) = P (4)

∑3i=1[q(i|4)] + d(4)

b(3)(4.15)

About state 4

P (3)b(3) = P (4)[d(4) +3∑i=1

q(i|4)]

P (4) = P (4) (4.16)

A similar rule to above can be used to calculate these probabilities. At this point a

pattern is starting to emerge but is still not absolutely certain, hence a five state Markov

Chain was also needed to hopefully create a general case.

About state 1:

P (1)b(1) = P (2)d(2) +5∑j=2

P (j)q(1|j)

P (1) =P (2)d(2) +

∑5j=2 P (j)q(1|j)

b(1)(4.17)

About state 2

P (2)[d(2) + b(2) + q(1|2)] = P (1)b(1) + P (3)d(3) +5∑j=3

P (j)q(2|j)

= P (2)d(2) +5∑j=3

[P (j)q(1|j)] + P (3)d(3)

+5∑j=3

[P (j)q(2|j)

P (2) =

∑5j=3[P (j)q(1|j)] + P (3)d(3) +

∑5j=3[P (j)q(2|j)

b(2)

=P (3)d(3) +

∑2i=1(

∑5j=3[P (j)q(i|j)])

b(2)(4.18)

About state 3

P (3)[d(3) + b(3) +2∑i=1

q(i|3)] = P (2)b(2) + P (4)d(4) +5∑j=4

[P (j)q(3|j)]

= P (3)d(3) +2∑i=1

(5∑j=3

[P (j)q(i|j)]) +5∑j=4

[P (j)q(3|j)]

39

P (3)b(3) =3∑i=1

(5∑j=4

[P (j)q(i|j)]) + P (4)d(4)

P (3) =P (4)d(4) +

∑3i=1(

∑5j=4[P (j)q(i|j)])

b(3)(4.19)

About state 4

P (4)[d(4) + b(4) +3∑i=1

q(i|4)] = P (3)b(3) + P (5)d(5) + P (5)q(4|5)

= P (4)d(4) +3∑i=1

(5∑j=4

[P (j)q(i|j)]) + P (5)d(5)

+ P (5)q(4|5)

P (4)b(4) =4∑i=1

(5∑j=5

[P (j)q(i|j)]) + P (5)d(5)

P (4) =P (5)d(5) +

∑4i=1(

∑5j=5 P (j)q(i|j))

b(4)

(4.20)

In general form

P (a) =P (a+ 1)d(a+ 1) +

∑ai=1(

∑Kj=a+1 P (j)q(i|j))

b(a)(4.21)

Where i is less than j and K is the population cap.

Notice how similar Equation (4.21) is to Equation (2.21) except for the fact that Equa-

tion (4.21) has the summation on the top of the fraction relation to catastrophes, hence

if q = 0 then the two are the same equation effectively.

Solving Equation (4.21) using Appendix B.1 in MatLab, the results are Figures 2.4,

2.5 and 2.6 which is an extremely interesting result.

4.4 Discussion

In this chapter it has been shown that the use of catastrophes does not change the pop-

ulation distribution in any shape or form but has massive effects on the expected times

40

to extinction. The distributions not changing does make logical sense if the randomness

term follows normal characteristics of an error term, in that the average of a series of error

terms is zero, however with very large terms it may be seen that the standard deviation

may change ever so slightly (or else if the value of q was dramatically increased). The

expected times to extinction significantly dropping could be expected too because there

is a higher chance of dying and over several hundred time intervals these probabilities

add up eventually having a noticeable effect of the resultant graphs of times to extinction.

Now that a general form including catastrophes has been derived assuming a continuum

approach, the next step is to make sure the work done so far in this report is accurate.

The next Chapter looks into computer simulation to hopefully back up any calculations

done so far.

41

Chapter 5

Simulation with NetLogo

NetLogo is a computer simulation program that has a stochastic component which can

simulate some of the models created in previous chapters; namely the times to extinction

and population distributions. The results will then be linked to previous results and

compared to see if they back up any of the derivations.

The code created is located in Appendix A. Below in Figure 5.1 is a screenshot showing

NetLogo in the middle of a simulation.

In Figure 5.1, the big black square is where the simulation running is visible (wasn’t

needed for this), on the left of that is a load of sliders which alter the variables mentioned

before as: b; ; q; p; K. Below the sliders are just a few monitors to assess numbers at

any given time interval and on the right of everything else is a line graph plotting the

number of ’turtles’ currently in the system (i.e. state j at time t).

Time in NetLogo is measured as ’ticks’, this can be treated as a dummy variable and just

be considered as one time interval, i.e. assume to be a very small amount of real time.

42

Figure 5.1: A screenshot of NetLogo running a simulation in order to collect random data

with preset variables.

5.1 Code Derivation

In the procedure ’to birth’ the population had to be checked at every tick to make sure

its numbers weren’t over the cap before asking the turtles if they was going to reproduce

or not, the random float had to be to three decimal points because the catastrophe rate

is every 1 in a thousand and this kept numbers similar throughout the program, if the

population was below the cap (previously referred to as ’K’) then the turtles were asked

if they would reproduce and then each turtle was also asked individually.

The ’to death’ procedure works in the same way as birth but there was no check to

make sure the population was below the cap, in the ’to go’ procedure it checks the popu-

lation isn’t extinct before performing this procedure so no additional checks were needed

at all.

The procedure starting ’to catastrophe’ is two procedures in one. First, each turtle

is asked if a catastrophe has happened and the chance of the catastrophe occurring is

determined with a slider. Secondly, if there is a catastrophe then there is a chance that

the turtles dies. The procedure happens in this order to make sure that this elevated

43

death rate only occurs when a catastrophe has actually happened.

The last three procedures all act together to create a list of the total count of turtles

at the end of each tick and then create an external file where this data is saved for use

with other programs such as ’IBM SPSS Statistics 22’, which was used to create the

Histogram 5.2 (this will be discussed later).

5.2 Results

As with the rest of the project, not only have estimated times to extinction been included

but a population distribution is also needed to accurately estimate when there is a cap

on the population.

5.2.1 Expected Times to Extinction

The table 5.2.1 displays the settings the simulation started off with each time and the

results are on the right side of the double vertical line. All values on the left of the double

vertical line are probabilities.

The ’Birth’ column represents the probability that a turtles reproduces with each tick (a

tick is considered a small amount of discrete time in this case). The ’Death’ column is the

probability that each turtles dies within each tick, the ’Catas.’ column is the probability

that a catastrophe happens at each tick. The ’Catas. D.’ column is the probability that

the turtles dies given that there is a catastrophe happening in that tick. The ’Cap’ is the

population cap, it is used to create the initial amount of turtles along-side ’In. Dens.’

(initial density) and the population can never exceed the cap as long as it isn’t altered.

The ’Av. Turts.’ (average turtles) is the average turtles present at each tick over 50

separate simulations. The ’Mean’ is the average time to extinction over 50 individual

simulations and the ’Sta. Dev.’ is the average standard deviation over 50 simulations.

44

Table 5.1: Table of Simulation Slider Settings and Results

Birth Death Catas. Catas. D. Cap In. Dens. Av. Turts. Mean Sta. Dev.

0.1 0.02 0.001 0.2 10 0.5 2 132.2115385 118.1651116

0.5 0.02 0.001 0.2 10 0.5 3 4554.173077 3847.983966

0.1 0.1 0.001 0.2 10 0.5 1 13.01923077 10.92665456

0.1 0.02 0.4 0.2 10 0.5 3 21.82692308 19.36615316

0.1 0.02 0.001 1 10 0.5 1 114.2884615 94.70083681

0.1 0.02 0.001 0.2 20 1 2 161.4807692 120.576148

0.1 0.02 0.001 0.2 10 0.25 2 101.4615385 100.6339633

0.1 0.02 0 0.2 10 0.5 2 149.2307692 124.3965221

The top set of data is the standard set of data to which the rest of the sets of data will

be compared to, this is also the same data that was used to derive the Markov chains

and equations graphs in previous chapters when the cap (K) was equal to ten.

The second set of data has an increased probability of birth and while it was expected

that the mean survival time would increase, the size of increase is surprising, where the

chances of birth is 500% the mean has risen over 10 times what it was before.

The third data set sees an increase in the probability of death and has the expected

result of decreasing the expected time to extinction.

This fourth set of data is key to what this project is looking in to, here there is a massive

increase in the probability that a catastrophe occurs and it has drastically decreased the

expected time to extinction. The results at this stage have back up further any work

done in previous chapters as well as backing up a common sense approach.

45

The sixth set of data, a couple of variables have changed but between the two the only

difference is that the simulation starts of with 20 turtles instead of 10, this increased the

mean a little bit but not by a lot. This result may be linked to the graphs previously

produced of expected times to extinction where as the numbers approached the cap the

expected time to extinction had very little change. Also, comparing the average turtles

to the cap, the average number of turtles rarely gets anywhere near close to the cap which

may explain the very small increases in time as the population reaches its maximum.

Again the penultimate data set is playing with the number of turtles starting the simu-

lation off, and again desired effect but nothing extraordinary.

The last data set is also very interesting as it investigates Chapter 3 where there was

no catastrophes present but the expected times to extinction didn’t jump as much as

hypothesised when generalising from previous findings.

5.2.2 Population Distribution

To create a population distribution a list at the end of each simulation was created where

the number of turtles was recorded at the end of each tick and then SPSS was used to

create the Histogram 5.2. This histogram was created with the 50 simulations of the

original data set and even though the cap is 10, looking at the diagram it is possible to

see that the population never even reached 5.

5.3 Discussion

So now there are some numerical values which solidify the results so far in previous chap-

ters. It has been shown that including catastrophes does not have much effect in terms of

average turtles present at any time interval t. However, including catastrophes does have

a noticeably significant difference on average times to extinction of the population as a

46

Figure 5.2: Poisson Distribution Histogram Using Data from NetLogo. Data collected

when K=20, d=0.02, b=0.2, q=0.001, p=0.2 and starting population is also 20.

whole. With NetLogo it has also been possible to demonstrate how much catastrophe

rates and birth/death rates have as an effect on the system without having to do scientific

study in real life.

While these simulations have been a very good numerical display of the report’s re-

sults so far, looking into Appendix A it is possible to see that there is several elements

of randomness. Chapter 6 looks into the possible effects of random errors and tries to

analyse them in a way that a random path can then be predicted if the different rates

and variables are previously known.

47

Chapter 6

Brownian Motion

Brownian motion is used to turn discrete data into continuous data that still includes a

random error terms which can then be differentiated and integrated etc. to be analysed

on a more general basis.

6.1 Fokker-Planck Equation

Using a similar approach to that of D. Ludwig [2] by tarting off with a deterministic

equation:

dx

dt= b(x)

dx = b(x)dt (6.1)

From here a random error term is needed to analyse any random events in the natural

world, hence adding to Equation (6.1):

dx = b(x)dt+√εa(x)dW (6.2)

The above equation is known as the Ito Equation where epsilon is small. If there is no

noise then a(x) = 0 and the term introducing the random error disappears and it just

returns to a deterministic approach like in Equation (6.1). Also given that X(t)=x,

b(x) = limt→0

1

∆tE[X(t+ ∆t)−X(t)]

εa(x) = limt→0

1

∆tE[(x(t+ ∆t)−X(t))2] (6.3)

48

If X was plotted against t as t was increased along the horizontal axis using Equation

(6.1) then the graph would jump in discrete steps. Let v be equal to the sample path of

these graphs and treat it as continuous for all t then using Equations (6.2) ... (6.3),

δv

δt=

ε

2

δ2

δx2(a(x)v)− δ

δx(b(x)v) (6.4)

Equation 6.4 is known as the Fokker-Planck Equation.

6.2 Sample Path Equation Derivation

Starting off with the logistic equation (Equation 6.5):

dx

dt= rx

(1− x

K

)(6.5)

From here a substitution is made where t = rτK→ 1

τ= r

tK. Then it follows,

d

dτ=

r

K

d

dt(6.6)

Then subbing Equation (6.6) back in to Equation (6.5) then follows,

r

K

dx

dt= rx

(1− x

K

)1

K

dx

dt=

x

K(K − x)

x = x(K − x) (6.7)

With this done, it is still currently a deterministic equation so an element of noise is

added. Since x and t are parametres that we cannot alter,

K = 1 +√εdW

dt(6.8)

From here, substituting Equation 6.8 back into Equation (6.7) then it follows:

x = x

(1 +√εdW

dt− x

)dx

dt= x+ x

√εdW

dt− x2

= x(1− x) + x√εdW

dt

dx = x(1− x)dt+ x√εdW (6.9)

49

Where W is a sample path function including normalised Brownian Motion. Now com-

paring Equations (6.4) and (6.9) we get:

dtv =ε

2d2x(x

2v)− dx(x[1− x]v) (6.10)

By setting dtv = 0 and w = x2v in Equation (6.10) and then integrating the following

happens:

0 =ε

2d2x(w)− dx(x[1− x]v)∫

0dx =ε

2

∫wxxdx−

∫ d

dx(x[1− x]v)dx

0 =ε

2wx − x(1− x)v

=ε

2wx − xv − w

=ε

2wx −

w

x− w

=ε

2wx −

1− xx

w (6.11)

Now re-arranging and integrating for a second time it follows:

0 = wx −2

ε

1− xx

w

dw

dx=

2

ε

1− xx

w

1

wdw =

2

ε

1− xx

dx∫ 1

wdw =

2

ε

∫ 1− xx

dx

ln(w) + c =2

ε

∫ 1

x− 1dx

(6.12)

Where c and k are integration constants from each side.

=2

ε(ln(x)− x+ k)

eln(w)+c = exp(2

ε(ln(x)− x+ k)

cw = exp(2

ε(ln(x)− x+ k) (6.13)

Where c = ec. Where c = 1c

and recalling w = x2v, Equation (6.13) can be rearranged

to:

w = c ∗ exp(2

ε[ln(x)− x+ k])

50

v =c

x2exp(

2

ε[ln(x)− x+ k]) (6.14)

Referring back to the Logistic Equation, there are two stationary points where x = 0

is an unstable stationary point and x = 1 is a stable stationary point when the system

is deterministic. The sample path Equation (6.14) of the logistic model is where there

is an element of noise also present, hence the error term (ε). However, there is still an

unknown variable k that has to be determined.

Thinking about the logistic equation distribution graph: If the error term is small (i.e.

Figure 6.1: A Graph to show the typical distribution of the deterministic Logistic Equa-

tion.

|ε| < 1) then the distribution of Equation (6.14) will look something similar to Figure

6.2. From this we can see that dvdx

= 0 at x = 1. So it follows:

dv

dx= − 2c

x3exp

(2

ε[ln(x)− (x− k)]

)+

c

x2exp

(2

ε[ln(x)− (x− k)]

)(2

ε

[1

x− 1

])(6.15)

Now substituting in what we know about vx then Equation (6.15) becomes:

0 = −2ce2ε(k−1) (6.16)

51

From Equation either c = 0 which implies v = 0 or exp(

2ε(k − 1)

)= 0 for very small ε.

Considering ex being approximated in Taylor form:

ex = 1 + x+x2

2!+x3

3!... (6.17)

Then,

e(2ε(k−1)) = 1 +

2

ε(k − 1) +

(2ε

)2(k − 1)2

2!+

(2ε

)3(k − 1)3

3!+ ... (6.18)

When ε is small, (2ε)n very quickly dominates and tends away from 0, unless k − 1 tends

to 0 faster. From this if k = 1 then Equation (6.18) tends back towards 1 which is as

close to 0 as possible. Substituting this value of k back into the sample path equation

then:

v =c

x2exp

(2

ε[ln(x)− (x− 1)]

)(6.19)

Figures 6.2, 6.3 and 6.4 where all solved with the fixed value of c = 1. They all carry

6.3 Numerical Analysis

Figure 6.2: A Graph Plotting Equation (6.19) by using the Maple Code in Appendix C.1

where ε = 0.1, k = 1.

a similar shape compared to the typical distribution graph of thee logistic equation, the

52


where ε = 0.01, k = 1.


where ε = 0.001, k = 1.

main aim was to get the centre of the distribution around x = 1. It is clear to see that

in the last three graphs that as epsilon is increased the wider the bell curve becomes,

as well as when epsilon approaches 0 then the bell curve becomes very sharp very quickly.

53

As the error term is increased it has been observed now that the distribution spreads

a lot more evenly, compared to a sharp spike at x = 1, this is to be expected as now the

element of noise can act as a measure of confidence for the collected data. The graph

spreads out more as epsilon is increased because our error term is the measure of spread

in the data.

This spread in data means that confidence intervals can be introduced with a bit more

work, choosing a suitable value of ε can give a suitable representation of the collected

data and confidence limits can be associated with the estimates for times to extinction.

Below in Figures 6.5 and 6.6 are still plotted with a fixed value of c = 1 but this time k

is varied instead of ε. For good measure, it is possible to see that even with very small


where ε = 0.1, k = 0.

changes in k, if it doesn’t equal 0 then the distribution graph for v very quickly tends to

0 if k < 1 and very rapidly shoots off to infinity if k > 1. Therefore keeping k = 1 makes

logical sense to keep the distribution graph in good proportion.

54


where ε = 0.1, k = 2.

6.4 Discussion

With some work starting from Equation (6.5) it is now possible to predict a more accurate

population distribution that includes the error that would occur in real life situations.

This method was able to be used with the Sample Path Equation as the rest of the work

in this report were based on the same assumptions and equations.

Looking at the graphs generated in Maple 18 with Appendix C.1 it is possible to see

some of the effects of changing the variables in Equation (6.19). Changing the error term

ε spreads the data out more and the bell curve in the middle therefore becomes wider as

ε is increased. As ε→ 0 then the line of the graphs slower approach to form the equation

x = 1, and as ε → inf the line slowly straightens out as v approaches cx2

. These results

make a lot of sense considering the Equation (6.19).

With a little work k was found to be the most applicable to the situation when it equaled

1, this kept the population distribution graphs in a reasonable and realistic perspective.

The fact that x = 1 is a stationary point of the Logistic Equation could also play a role

55

and with some further work may be able to be shown. A similar result was found with

fixing c to be 1 as well, but trying other values didn’t quite have the same dramatic effect

on the graphs as the value of k.

56

Chapter 7

Conclusions

Having looked at discrete data and a continuum approach, whilst also considering de-

terministic and stochastic approaches all the data fits together quite nicely. From the

story of the heath hens in Chapter 1 it is possible to see that there were a lot of different

occasions (here we called them catastrophes) where the population numbers were drasti-

cally decreased. Having looked into further detail on the classification of a certain species

survival chances the information was found that all animals are assessed on 5 criteria.

The criteria the Red List use that was the main focus in this paper was the ’probability

of extinction’. While the probability of extinction isn’t too difficult to calculate or gather

data for its a very deterministic variable that needs some element of noise, or confidence

limits, to justify the estimations made.

The discrete data used in Chapter 2 was a very basic technique but set guidelines to

how later on it was possible to do the same thing but with a continuum approach. In

Chapter 2 however the probability distributions were created for unique values of K which

proved to be consistent throughout the project and never had any visual differences, even

when new variables were thrown into the equation. These distributions were based on

work from Markov Chains. Even as the Markov Chains became increasingly messy and

difficult to work with the results almost identical. Random simulation with NetLogo

only backed this up further and that was also including an element of randomness. These

57

results were very confidence boosting in that it all backs each other up and therefore

has a high external validity. The equation in general form in Equation (4.21) was very

promising to find, while it looks basic at the moment it only considers constant rates

of catastrophes etc. If functions were put in there place here then the birth rates could

be more realistic (e.g. young babies aren’t going to be giving birth but constant rate

suggests so at the moment), functions to consider would be: the catastrophe rate; the

birth rate; the death rate; and the probability of dying in a catastrophe may be a simple

function or it may be completely determined by the type of catastrophe at hand. Another

thing to look at would be different types of catastrophe, while famine may take longer

to have effects its death rates during the catastrophe may be significantly larger than an

epidemic for example.

The next thing to consider is the expected times to extinction. In Chapter 2 these

made no sense at all where the expected times to extinction oscillated around above and

below 0, suggesting the population would become extinct and then suddenly be present

again. Assuming a continuum approach however soon rectified this giving some very

realistic looking graphs in Figures 3.1...3.3. These results showed characteristics of car-

rying capacity and disperse populations when the population numbers were approaching

0. The general shape of these graphs retained a general shape throughout the paper but

catastrophes had a very heavy effect on the vertical axis representing the times them-

selves, where the values all dropped around 120 time units. At this stage the data was

still assuming a deterministic approach and soon changed a little bit again when an error

term was introduced. However the shape again held constant throughout.

Looking at the birth and death rates these had the expected results as anyone could

assume from the start, the higher the value of b(j) the longer the population would last,

on average, this would be because the members of the population would be producing

more offspring in their lifetimes and raising d(j) would have the opposite effect by giving

the population less time to reproduce. The catastrophe rate had very small implications

58

in the computer simulation compared to them predicted in previous chapters. There is

still a lot of work to do with these rates as different species have different mortality rates

at different ages (which is another factor to include with further work) such as in general

the older the population becomes the more likely those members are to die. Also, as stated

before very young offspring will have an extremely low reproductive rate and this will rise

in times of maturity and drop once again in old age. Catastrophe rates will determine on

the catastrophe at hand, for example if natural disasters are being considered then there

could be a difference related to geographic location or if disease is at hand is one species

more susceptible compared to others that it causes death in some but not others, also is

the disease prominent in some areas compared to the other side of the planet for example.

In the previous chapter, a lot of work based on D. Ludwig, went into observing the

effects of ε on the sample path equation (Equation (6.19)) and the results were shown to

be as expected. Increasing the value of ε made the distribution curves noticeably wider

with every factor of 10 that ε was increased by; eventually if the error term is too large

where it becomes insignificant then the curve starts to flatten out and the data would

then be very difficult to analyse. Some further work may be done here to find a more

reliable proof for setting the value of k to be 1, and also to see how these results compare

to some of the deterministic approaches discussed in earlier chapters as ε→ 0. It would

also be interesting to see if the error term could be introduced somewhere to Equation

(4.21) and compare to the work done in Chapter 6.

With a bit more work, the next step here would be to look into moment generating

formulae (similar to that of Mangel and Tier [7]) where the deterministic and stochastic

equations could be interpreted with this approach and once again the results compared

in more detail. The use of moment generating formulae could also bring about the in-

troduction of standard deviation, which would lead to the confidence limits suggested in

the introduction alongside the expected time to extinction as a replacement to the Red

List criteria currently in place (probability of extinction).

59

At the moment the work done here could make some estimates into the expected time to

extinction up to about 50 years without any problems, and maybe even 100 years if the

variables remained similar to the present time. This could be a very powerful tool and

with a bit more work could possibly be a very efficient replacement of some of the Red

List’s 5 criteria. Trying to use the material in this paper to generalise to populations

that are vastly widespread across the planet (e.g. humans) would be very difficult as the

work here is based on populations that are very specific in area. Like it was suggested

earlier, generalising to wider areas could be introduced if a form of geographic location

was somehow introduced depending on the area; for example humans can survive a lot

easier in the United Kingdom compared to the middle of a desert.The introduction of

so many factors however would complicate calculations very quickly and could result in

a lot more data being needed to eventually take an average of several expected values

overall.

60

Bibliography

[1] Image View of the Different State of the Red List.

http : //commons.wikimedia.org/wiki/F ile : Status iucn3.1.svg

(Accessed: 20th October 2014).

[2] Donald Ludwig (1975)

Persistence of Dynamical Systems Under Random Perturbations

University of British Columbia

[3] The Red List and how Species are Catagorised.

http : //education.nationalgeographic.com/education/encyclopedia/endangered −

species/?ar a = 1

(Accessed: 18th Jan 2015).

[4] The Heath Hen on Martha’s Vineyard.

http : //www.bagheera.com/inthewild/ext heathhen.htm

(Accessed: 23rd October 2014).

[5] Classification of Species Vulnerability.

http : //education.nationalgeographic.com/education/encyclopedia/endangered −

species/?ar a = 1

(Accessed: 19th Jan 2015).

[6] B. J. Cairns and P.K. Pollett (2005)

Approximating Persistence in a General Class of Population Processes

University of Queensland

61

[7] Marc Mangel and Charles Tier

A Simple Direct Method for Finding Persistence Times of Populations and Applica-

tion to Conservation Problems

University of California

62

Appendix A

NetLogo Code

A.1 Initial Simulation.nlogo

; variables

globals

[

current-turtles

all-turtles-ever

average-turtles

]

; the set up

to setup

clear-all

set all-turtles-ever ( 0 )

setup-turtles

update-turtle-globals

reset-ticks

end

63

to setup-turtles

create-turtles initial-density * cap

ask turtles

[

setxy random-xcor random-ycor

]

end

; updating globals

to update-turtle-globals

set current-turtles ( count turtles )

set all-turtles-ever (all-turtles-ever + count turtles )

end

; main procedure

to go

if count turtles = 0

[

stop

]

move-turtles

birth

death

catastrophe

update-turtle-globals

tick

export-files

end

64

to move-turtles

ask turtles

[

right random 360

forward 1

]

end

to birth

ask turtles

[

if count turtles < cap

[

if random-float 1.000 < probability-birth * count turtles

[

hatch 1

]

]

]

end

to death

ask turtles

[

if random-float 1.000 < probability-death * (count turtles)*(count turtles)

[

die

]

]

65

end

to catastrophe

ask turtles

[

if random-float 1.000 < probability-catastrophe

[

if random-float 1.000 < probability-death-catastrophe [ die ]

]

]

end

to export-data

;;set the directory where the file will be stored

set-current-directory "C:\\Users\\Ben\\Documents"

;; create the column headings once

if ticks = 1 [create-files]

export-files

end

to create-files

;; create the file and give the first row column headings

file-open "Turtle-Tick Count.doc"

file-print (list "total")

file-close

end

to export-files

;; write the information to the file

66

file-open "Turtle-Tick Count.doc"

ask turtles [file-print (list count turtles)]

file-close

end

67

Appendix B

MatLab Code

B.1 Dissertation.m

%Defining ’Independent’ Variables

K = 10;

b = 0.1;

d = 0.02;

%Defining ’Base’ Matricies

a = eye(K,K);

c = zeros(K,K);

B = ones(K,1);

for j=1:K-1,

a(j,j+1) = -b*j;

a(j+1,j) = -d*j^2;

end

A = inv(a);

68

C = zeros(K,1);

P(1)= 0.0069595;

for j = 1:K

P(j+1) = (b*j*P(j))/(d*j^2);

end

for j = 1:K

C(j,1) = P(j);

end

Z=0;

for j = 1:K

Z = Z+ P(j)*j;

end

p = 0.2;

q = 0.001;

for j=1:K

c(j,j) = -(b*j + d*j*j + q + q*(1-p)^j);

end

c(K,K)=-d*K*K;

for j=1:K-1

c(j,j+1) = b*j;

c(j+1,j) = d*(j+1)*(j+1) + q*j*p*(1-p)^(j-1);

end

m_one = ones(K,1);

for n=1:K

for k=1:K

if n - k >= 2

69

r = nchoosek(n,k);

c(n,k) = q*r*p^(n-k)*(1-p)^k;

end

end

end

A*m_one

plot(A*m_one)

B.2 Binomial.m

% Using markov chains to obtain a distribution (with catastrophes)

K = 10;

P=zeros(K,1); P(K) = 0.000000000000013408/1.8844e-012/1.0682/1.0186/0.9312/2.0732;

d = 0.02;

b = 0.1;

q = 1;

p = 0.5;

for a = K - 1: -1: 1

P_(a) = P(a + 1) * d * (a + 1)^2 + Binomial_Internal(a, K, P, p, q);

P(a) = P_(a) / (b * a);

end

% A seperate function to work out the summation on the top of the

% fraction used to calculate the probabilty distribution for

% markov chains with catastrophes.

70

function sumR = Binomial_Internal(a, K, P, p, q)

%P = P(K)

R = zeros(K,1);

for i = 1: a

% R(K, 1) = P * q * (nchoosek(K, i) * p^(K - i) * (1 - p)^(i));

for j = a+1:K

R(i) = P(j) *q* (nchoosek(j, i) * p^(j-i) * (1 - p)^(i));

end

end

sumR = sum(R);

B.3 Continuum Matrix.m

% Defining indepedent variables

K = 10;

b = 0.1;

d = 0.02;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% For a Continuum Approach

C = zeros(K,1);

P(1)= 0.0069595;

p = 0.2;

q = 0.001;

71

for j=1:K

c(j,j) = -(b*j + d*j*j + q + q*(1-p)^j);

end

c(K,K)=-d*K*K;

for j=1:K-1

c(j,j+1) = b*j;

c(j+1,j) = d*(j+1)*(j+1) + q*j*p*(1-p)^(j-1);

for i=1:j-2

c(j,i) = i*nchoosek(j,i)*p^(j-i)*(1*p)^i;

end

end

minusone = (-1)*ones(K,1);

for n=1:K

for k=1:K

if n - k >= 2

r = nchoosek(n,k);

c(n,k) = q*r*p^(n-k)*(1-p)^k;

end

end

end

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

plot(c\minusone)

72

Appendix C

Maple Code

C.1 V Graphs.mw

c := 1:

epsilon := 0.1:

k := 1:

v := (c/(x^2))*exp((2/epsilon)*(ln(x)-(x-k))):

plot(v,x=0..2);

73