Download Fundamental Models in Population Biology – Cribs

Part II Zoology – M3 Population Biology
Fundamental Models in Population Biology – Cribs
1. a) Lotka-Volterra predator-prey model
To do:
Model A
 Turn Model A “On” under the “Model Selection” section in the top-left of the screen.
1. Move the “a” slider to vary the hare birth rate.
a. How does this influence the hare and lynx population sizes?
b. Why do you think this happens?
Model B
 Turn Model B “On” under the “Model Selection” section in the top-left of the screen.
2. Observe the new behaviour after the addition of bounded prey growth.
a. How does the introduction of intraspecific density dependence change the dynamics of the model, i.e.
how does the behaviour in model B differ from model A?
3. Play with all of the parameter sliders to explore the changes in the dynamics.
a. Give a set of parameter values for which the two populations cannot coexist (ie. there is no positive
non-trivial equilibrium)..
b. In this case, where does the population tend to?
c. Give a set of parameter values which results in the lynx population dying out.
d. Slowly vary each of the parameter sliders and observe which parameters have a significant effect on
returning the system to coexistence.
Model A
1a.
 Increasing the hare birth rate increases the frequency and amplitude of the oscillations of both
populations.
 The hare population peaks before the lynx population, irrespective of the value of a.
 At low values of a, the peak of the hare population is greater than that of the lynx. At high values of a,
the reverse is true.
1b.
 Increasing the hare birth rate will increase prey availability for the lynx, which will consequently increase
the lynx population.
 The delay arises from the time taken for the lynx to convert the extra prey into offspring.
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Part II Zoology – M3 Population Biology

The coupled oscillations arise from when the lynx population reaches a critical number at which the
predation rate on the hare causes a decline. The then reduced hare population will cause the lynx
population to decline from reduced prey availability.
Model B
2a.
 The addition of bounded prey growth stabilises the system.
 Both populations oscillate towards the stable equilibrium eventually fixing at the stable population sizes.
 Looking at the phase plane, in model A the populations circle about the stable equilibrium, whilst in
model B the populations spiral in towards the stable equilibrium. Technically, the equilibrium in model
A is a neutral focus, and therefore the system is structurally unstable, whilst in model B it is a stable
focus rendering the system stable.
3a.
 Set m and k to very low and n to high and no positive non-trivial equilibrium exists, irrespective of the
values of the other two parameters.
3b.
 The lynx population dies out and the hare population tends to a stable size of about 50.
3c.
 As before.
3d.
 A low carrying capacity of hares, high death rate of lynx and low conversion of consumed hares to new
lynx does not allow for the two species to coexist. Importantly, this corresponds to low prey availability
and then low efficiency of energy retrieval from consumed prey, which in combination means that the
lynx population cannot persist. The key parameters in promoting coexistence are therefore m and k.
 The a and b parameters have no significant effect on returning the system to coexistence. The lynx
death rate, n, must be reduced to low values to achieve coexistence.
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Part II Zoology – M3 Population Biology
SUMMARY: Lotka-Volterra predator-prey model
Application: Model a predator-prey interaction.

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In the absence of predators, the prey population will grow exponentially (model A) or will grow to
the carrying capacity (model B).
In the absence of prey, the predator population will decline exponentially.
Encounters between predators and prey are in accordance with mass-action.
- The rate of prey removal as a consequence of an encounter is determined by parameter
b.
- The rate of new predators born as a consequence of an encounter is determined by the
parameter m.
In model A, coupled oscillations are observed.
- When prey numbers are high, predators increase due to increased prey availability but
this then imposes increased predatory pressure on the prey which causes prey numbers
to decline. When prey numbers are low, predators will decrease from decreased prey
availability consequently creating these coupled oscillations.
- These oscillations would carry on indefinitely if the system experienced no external
disturbances; this is due to the equilibrium being neutrally stable.
In model B, both populations oscillate before stabilising at the stable population sizes.
- The concept of the carrying capacity limits the prey population size due to intraspecific
density dependence.
- The addition of bounded prey growth caused by intraspecific density dependence
stabilises the system.
- The equilibrium is now a stable focus and provides fixed population sizes at which the
two species can coexist.
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Part II Zoology – M3 Population Biology
1. b) Lotka-Volterra competition model
To do:
 Double click on “LotkaCompetition.m” in the “Current Folder” window and run the code.
1. Consider the phase portrait.
a. Can the long term dynamics of the system be influenced by adjusting the initial numbers of individuals
in each population?
b. Estimate the range of initial conditions where M. digitata dominates.
c. Check your estimate by altering the values in the initial population size boxes.
2. Vary the effect of interspecific competition on both species of wasp by moving the “
” and “ ” sliders.
Keep K1 and K2 at the default values 114.8 and 86.8, respectively.
a. Use the phase portrait to estimate the level of interspecific competition required for N. vitripennis to
always outcompete M. digitata, irrespective of the initial population sizes?
b. Repeat for when M. digitata outcompete N. vitripennis.
3. Set
and
to a large value so that the situation where interspecific competition has a large effect on both
species can be considered.
a. Vary the effect of intraspecific competition on both species by altering K 1 and K2, and study the phase
portrait. Persuade yourself that it is hard for coexistence to exist.
b. Why does the level of interspecific competition influence coexistence so prominently?
1a.
 Yes, there are two stable equilibria with this combination of parameter values and so the starting
population sizes determine which of these stable equilibria are reached.
 Keeping all of the default parameter values, all starting population sizes to the left of the diagonal line
through (0, 0) and about (30, 20) will go to the equilibrium where N2 persists, and all to the right will go
to the equilibrium where N1 persists.
1b and c.
M. digitata will dominate when
the initial conditions fall on this
side of the diagonal line.
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Part II Zoology – M3 Population Biology
2a.
 Set
high and
low.
2b.
 Set
low and
high.
 One species will outcompete the other in all cases when only one positive stable equilibrium exists.
 When one species has a strong interspecific competitive ability, this reduces the likelihood of species
coexistence.
3a.
3b.
 As before, strong interspecific competitive ability reduces species coexistence. In this case, both species
competes with the other very strongly creating a situation where coexistence is very unlikely to be
reached.
 By considering null clines and the stability of the equilibria, the following conditions can be derived for
coexistence:
From this, it is evident that as the interspecific competitive ability of one or both of the species is
increased, the less likely coexistence is to exist.
SUMMARY: Lotka-Volterra competition model
Application: Model two populations competing for the same resource.
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Based on the Lotka-Volterra, but now with both intra and interspecific density dependence.
The inclusion of the competitive effect of each population on the other provides the basis for
modelling the dynamical effect of competition from another population.
- When
, the interspecific effect is less inhibitory than the intraspecific effect on
population 1.
- When
, the interspecific effect is more inhibitory than the intraspecific effect on
population 1.
Four outcomes can exist dependent on the combination of parameter values:
- Population 1 always excludes population 2.
- Population 2 always excludes population 1.
- One population will exclude the other dependent on initial population sizes.
- Both populations reach a stable coexistence.
Coexistence only exists when both populations have a greater intraspecific inhibitory effect than
interspecific effect; that is, they compete more with themselves than with the other population.
Again, intraspecific density dependence has proved to have a stabilising effect.
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Part II Zoology – M3 Population Biology
2. Nicholson-Bailey model
To do:
 Double click on “NicholsonBailey.m” in the “Current Folder” window and run the code.
1. Consider a scenario when the level of aggregation is high by setting k to a very small value.
a. Slowly increase k up to
and comment on how increasing aggregation influences the time it
takes for the populations to reach stability.
b. How does the long-term outcome of the populations change if
?
2. Set k to approximately 0.8 thus assuming some aggregation.
a. Slowly decrease the birth rate of the whiteflies by moving the “b” slider and comment on how this
affects the dynamics of both populations.
3. Set k to approximately 0.4, thus assuming a high level of aggregation, and set b to approximately 2.
a. Does varying the initial population sizes have any effect on the eventual stable population sizes?
1a.
 The population dynamics exhibit damped oscillations tending towards the equilibria.
 The population oscillates increasingly more as k is increased and therefore it takes longer for stability to
be reached.
1b.
 The population continues to oscillate at increasingly higher amplitudes as time increases.
 As k gets larger, the population exhibits much more significant peaks and troughs to the point that the
population will in reality die out as a population of 0 is reached.
 The population never stabilises for
.
2a.
 The dynamics are similar but the stable population size for both species declines.
 At birth rates close to 1, the oscillations become slower and larger.
3a.
 There is no effect on the stable population sizes for both species from varying the initial population
sizes, when all parameters remain constant.
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Part II Zoology – M3 Population Biology
SUMMARY: Nicholson-Bailey model
Application: Model host-parasitoid interactions (in this case: accounting for degrees of
aggregation).
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A host that encounters a parasitoid becomes infected and therefore yield new parasitoids whereas
a host that is not encountered is assumed to reproduce.
Encounters are assumed not to be random in the heterogeneous version of the Nicholson-Bailey
model, but aggregated; this is represented by the introduction of the k parameter.
- Maximal aggregation when
.
- Random encounters and therefore retrieval of the original Nicholson-Bailey when
.
A significant level of aggregation (
) stabilises the system.
- Increasing aggregation increases stability.
- Decreasing aggregation increases oscillations and, when
, the system becomes
unstable exhibiting behaviour more similar to the original Nicholson-Bailey model.
Increasing aggregation increases stability through the increase in direct density dependent effects.
- At high parasitoid densities and with high aggregation, a parasitoid aggregation is more
likely to come across host patches where most or all of the hosts are already parasitized.
- This will therefore decrease the effective attack rate of the parasitoids and consequently
their birth rate dampening the population oscillations and reducing the parasitoid influence
on host mortality.
- Parasitoid efficiency declines as parasitoid density increases; this trade-off presents
stability.
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Part II Zoology – M3 Population Biology
3. SIR model
To do:
 Double click on “SIR.m” in the “Current Folder” window and run the code.
1. Keep the default values fixed whilst increasing the recovery rate.
a. How does this affect the number of boys that caught influenza?
b. Explain why this is the case.
 Return to the default values.
2. Suppose the school was smaller, try incrementally decreasing the population in the “Total number of boys”
box. Remember to take the new N and then enter N-1 into the “Number initially susceptible” box.
a. How does a smaller school size influence the proportion of boys that became infected? (Note: the
graph axis scale will alter according to the initial conditions).
 Return to the default values.
3. Consider the phase portrait whilst varying the birth/death rate and recovery rate (b and ν sliders).
a. How do both of these influence the equilibrium value?
 Return to the default values.
4. Increase the rate of recovery to approximately 1. Suppose that a number of boys returned to school infected.
Vary the “Number initially infected”, remembering that the “Number initially susceptible” is the new number
of infected subtracted from the total number of boys.
a. How does the increased number of initially infected boys affect the proportion of boys that become
infected?
 Return to the default values.
5. Suppose we are now considering the dynamics of the infection over a long period of time and therefore births
are significant.
a. How does increasing the birth rate affect the long term dynamics of the disease?
Fix the birth rate at approximately 0.08.
b. Estimate the proportion of the population that you would need to vaccinate for the disease to be
eradicated.
c. How does varying the birth rate affect this proportion?
1a.
 Increasing the recovery rate decreases the number of boys that became infected and are therefore now
recovered.
1b.
 Recall that the infectious period is 1/ν and so this term decreases as ν increases.
 Increasing the recovery rate decreases the infectious period therefore decreasing the number of
encounters that infectious person can have with a susceptible person.
2a.
 The proportion of boys that become infected (i.e. the stable number of recovereds) decreases as the
population is decreased.
 Decreasing the total population decreases the likelihood of encounters being made between infecteds
and susceptibles.
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Part II Zoology – M3 Population Biology

Recall that the reproductive ratio (
) is dependent on the population size, and at a population
size of approximately 209
. If the school had a much smaller population, influenza may not have
been able to have sustained an epidemic (given fixed values for and ).
3a.
 Increasing the rate of recovery increases the equilibrium for the number of susceptibles, whilst
infectious remain at 0.
 Increasing the birth/death rate increases the equilibrium for both susceptible and infectious allowing
both to stabilise above 0.
4a.
 As the number of initially infected increases, the number of boys that become infected increases.
 More infected at one time will increase the number of encounters between susceptibles and infecteds.
5a.
 The numbers within each compartment fluctuate more as the birth rate is increased and the equilibrium
numbers are altered to be closer to one another.
 There are increasingly more susceptibles and infectious and fewer recovereds.
5b.
 At approximately p = 0.7, the number of infectious declines to 0.
5c.
 Increasing the birth rate increases the turnover of the population therefore requiring a lower proportion
of individuals to be vaccinated at birth.
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Part II Zoology – M3 Population Biology
SUMMARY: SIR model
Application: Model the spread of an infectious disease through a closed population.
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An individual in a population is in one of three states: susceptible to a disease, infected with the
disease and recovered from the disease.
- Transmission is based on mass-action theory and therefore is dependent on random
encounters between infecteds and susceptibles.
The reproductive ratio, , is interpreted as being the mean number of secondary cases in a
population of entirely susceptible individuals.
-
.
-
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An epidemic is possible when
.
This can be used to calculate the critical population size that denotes the minimum
population size of susceptibles required for the epidemic to occur.
determines the initial rate of disease spread and the proportion of susceptibles that
will be eventually infected.
The proportion of individuals vaccinated at birth strongly influences the spread of the disease by
reducing the number of available susceptibles in the population to encounter.
- The critical vaccination, , threshold provides the minimum proportion of vaccinations
at birth required to prevent an epidemic.
-
.
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