BIOL377代写、c++，Python程序代写

日期：2023-04-20 09:23

Simulating interacting populations

BIOL377

There are multiple ways that we can go about simulating species population dynamics in

order to explore the consequences of changes to those species’ abiotic and biotic

environment. For example, we might wish to explore what happens when changing

temperatures alter species metabolism and decrease or increase growth rates, how a

reduction in available habitat changes equilibrium population sizes, or how the introduction

of an exotic species might disrupt the dynamics of an undisturbed community. Each of these

disturbances, among many others, are increasingly common across modern-day

communities; however, most to all are impossible or unethical to test within real-world

conditions, hence the use of mathematical and computational models to investigate them in

silico . We will start the Research Workshop by interrogating three different (but related)

modelling scenarios that have been the subject of considerable study within the ecological

literature.

1. How to make an in silico model-based prediction

Before we can study any specific model, we need to understand the ground rules for how we

might go about using a population-dynamics model to make a prediction. By and large, there

are four critical ingredients required to make a model-based prediction about the species in

community of interest:

1. We need a mathematical description of how the size of the different populations are

expected to change over time. Typically, this mathematical model will depend on

parameters, which are things we normally treat as unvarying (e.g., temperature,

habitat availability, etc), and on variables, which are things that we allow to change

over time (e.g., the sizes of the different populations).

2. We need to decide what values to assign to the different parameters. Sometimes we

can directly relate these to measurable quantities to “ground” our simulation in a

specific realistic scenario. At other times, we will see that what the values actually

mean (and how they relate to each other) is a bit more nuanced.

3. We need to decide what the community state is at the “start” of our simulation. When

using population-dynamics models, we are effectively trying to make a prediction

about how populations will change over time. But to measure how they have

changed, we need to know where they changed from .

4. We need to decide how far into the future we wish to simulate and make our

prediction. Sometimes we are interested in making a short-term prediction, for

example if we want to know whether a new disturbance will make populations go up

or down. At other times, we are interested in making a long-term prediction because

we want to determine whether the community will recover from disturbance over a

long time horizon or whether or not it will end up in a completely different state due to

altered internal feedbacks.

2. Logistic growth of a single population

To make things as straightforward as possible, we will start by studying one of the

mathematically simplest models for a population varying over time: logistic growth. In this

model, we have a population of a single species whose abundance tends to increase when it

has a small population size relative to its carrying capacity and to decrease when it has a

large population size relative to its carrying capacity. Mathematically, the model looks like

= 1 ?

( )

where the equation describes the growth rate of the population density N at any given

moment of time. We can think of growth rate as the amount of change in population density

divided by the time over which that density changes. This implies that negative growth rates

lead to decreases in N over time and positive growth rates lead to increases in N over time.

In this model, the parameter r is the maximum growth rate of species N and K is the carrying

capacity of species N . Note that if N is much smaller than K , the total growth rate will be

positive as long as r is positive; if N = K , the total growth rate is zero; and if N > K the total

growth rate is negative. Code to explore this model in R can be found on Learn in the file

logistic_growth.R.

Though this model is simple mathematically, it sets an intuitively useful baseline for what

might happen if we add additional species or additional phenomena to the mix. This is what

is shown in the default scenarios outlined in the R code that has been made available to you.

What to do

1. Work through the R code in logistic_growth.R. Make sure that you can run it from the

top to bottom, that it all makes sense, and add comments to it in the places you think

you’d like reminders of what does what.

2. The different “experiments” in the code allow you to see what happens if you vary the

values of parameters like r and K . If you want to conduct further experiments, try

changing just one parameter at a time before you change both together. Before you

simulate what happens upon changing a parameter, try to make a guess about what

should happen based on your biological intuition. For example, if you increase K , do

you expect population sizes to go up or to go down?

3. The model as written makes predictions about what the population should do in the

absence of any external disturbance, but we know that global change is doing plenty

of things that impact species directly (and not necessarily by changing the underlying

growth rate or carrying capacity). One way to “disturb” the population is to introduce

some sort of perturbation and see what happens after (i.e. check its resilience). See

if you can change the R code you started with to explore the following simulated

disturbances:

a. Imagine that this is a threatened species. At time t=100 , we translocate 40

new individuals to the population. We then allow the population to go back to

its intrinsic dynamics. What happens to the population overall?

b. Imagine that this is a commercially viable species. At time t=100 , we remove

23% of the population. We then allow the population to go back to its intrinsic

dynamics. Can the population recover?

c. Imagine that this is a commercially viable species that we wish to exploit

continuously. Starting at time t=100 , we wish to remove a constant number of

individuals from the population at all times. What is the largest number of

individuals that we can exploit and maintain a viable population?

3. Dynamics of a predator and prey

We will now move to THE classic continuous-time model for interacting predator and prey

species: Lotka-Volterra. Mathematically, the model looks like

where the first equation describes the growth rate of N and the second the growth rate of P .

The parameter r is the maximum growth rate of N , K is the carrying capacity of N , a is the

attack rate of P attacking N , e is the efficiency of P eating N , and s is mortality rate of P . All

of these parameters should always be given positive values.

In this model, we have two different populations, N and P , who vary over time with respect to

each other. If the population of P is zero, the population of N shows logistic growth as

described above. If the population of P is not zero, then the realized growth rate of N

decreases because of the interaction with P . Likewise, if the population of N is zero, then the

population of P experiences mortality; but, if the population of N is not zero, then the realized

growth rate of P increases because of the interaction with N .

An important feature we can see in the model is that the equation for each species has both

positive and negative contributions to overall growth. Without these, we can end up with

models that don’t make biological sense. For example, the predator should die as time goes

on when there are no available prey. Likewise, the prey population should not grow forever,

even in the absence of the predator. Lastly, note that the terms that capture the interaction

between species (i) feature both densities being multiplied together (so that the interaction

“vanishes” whenever any species is absent) and (ii) are “paired” between the two models so

that loss of density of prey corresponds to a proportional gain of density for the predator.

Code to explore this model in R can be found on Learn in the file lotka_volterra.R. Similar to

the model explored above, changes to the different parameter values can change the

dynamic behavior and hence biological predictions that it makes. This is what is shown in the

default scenarios that have also been outlined in the R code.

In the R code, we start from a scenario where the prey population has an extremely large

carrying capacity and hence would grow exponentially without the predator. Here the

Lotka-Volterra model shows what’s called a fixed point equilibrium such that the populations

don’t vary over time. If we start the population away from this equilibrium, we see that the

two species will oscillate back toward this equilibrium. What does this mean in terms of what

the Lotka-Volterra model predicts for the resilience of a set of populations that might be

subject to some sort of external disturbance?

We next explore two related experiments in the code where we now have logistic growth

because intraspecific competition is no longer zero. In the first such scenario, we decrease

the prey’s carrying capacity which ends up driving the predator to extinction. (How small a

carrying capacity will allow the predator to avoid extinction?) In the second scenario, we

decrease the prey’s carrying capacity AND increase the predator’s attack rate. Somewhat

paradoxically, this helps prevent the predator from going extinct, even though the equilibrium

abundances of both predator and prey are much lower than where we started. Do these

results make sense to you? Why should competition between prey make it harder for the

predator to avoid extinction? Why would increased attack rates make things more stable?

What to do

1. Work through the R code in lotka_volterra.R. Make sure that you can run it from the

top to bottom, that it all makes sense, and add comments to it in the places you think

you’d like reminders of what does what.

2. See how the output (as captured in the plots) changes if you vary the parameters. Try

changing just one of them at a time before you change more than one. Before you

change a parameter, try to make a guess about what should happen based on your

biological intuition.

3. See if you can change the R code to explore the following scenarios:

a. The third and fourth experiments hint at the fact that the rate of intraspecific

competition and the predator attack rate combine to determine the fate of the

prey. Can you think of other parameters that you could have changed to get

similar qualitative predictions? See if you can change other parameters and

get figures that look similar to those in the middle and bottom row. Then work

out whether or not you can provide a biological explanation for why that

makes sense.

b. Explore what happens if you add another prey or predator to the system

described by the fourth experiment (since it gives stable coexistence). How

many new parameters do you need to add to describe the additional

biology/ecology? Think about how you might make simplifying assumptions

about the “behavior” of the new species. For example, can you assume that

some things can be identical to the species already present? If this is a new,

exotic species, what has to happen in order for it to establish? Does the

answer depend on if you add a new predator instead of a new prey?

4. Saturating predator functional response

All mathematical models are, at best, a caricature of reality. In the Lotka-Volterra model,

there is one assumption that has classically been regarded as less reasonable than all

others: a non-saturating functional response. If we look at the term that captures the

predator-prey interaction, we can see that the per capita rate of predation will always grow

regardless of the density of prey available. In real-world terms, this means that if you went to

a buffet the amount you would eat would always be proportional to the food available,

whether the buffet table was two meters long of two kilometers long. Plenty of empirical

evidence indicates that this is rarely the case, and instead it is much more likely that

consumption eventually saturates at some level. To build this into our model, we can change

the term for the interaction (which appears in both equations), as follows and get what is

known as the Rosensweig-MacArthur model

Note that, instead of a single attack rate a , we now have two different parameters a and h

which control the rate of consumption. In the literature, a is defined as the maximum capture

rate and h is the “handling time” of species P . If the number of prey is very very large, the

predators will never consume more than a prey per capita, but if the number of prey is small

then the consumption rate will increase proportional to N (just like in the original

Lotka-Volterra model!). Code to explore this model in R can be found on Learn in the file

rosenzweig_macarthur.R.

Similar to the models explored above, changes to the different parameter values can change

the dynamic behavior and hence biological predictions that it makes. This is what is shown in

the default scenarios outlined in the R code that has been made available to you. We start

from a scenario where the prey and predator populations can both co-exist with well-defined

equilibrium population sizes. If we increase the carrying capacity of the host, however, we

can shift to a situation where the populations cycle over time.

What to do

1. Work through the R code in rosenzweig_macarthur.R. Make sure that you can run it

from the top to bottom, that it all makes sense, and add comments to it in the places

you think you’d like reminders of what does what.

2. The difference between the experiments should remind you very strongly of

something you’ve seen in the previous examples. Can you figure out why the results

are so similar? See if you can figure out how to prove your hypothesis.

3. Explore how the output (as captured in the plots) changes if you vary the parameters.

Try changing just one of them at a time before you change both together. Before you

change a parameter, try to make a guess about what should happen based on your

biological intuition.

4. See if you can change the R code to explore the following scenarios:

a. Imagine that the habitat of the prey is undergoing land-use change. Estimate

how much habitat can be removed (via changes in the prey’s carrying

capacity) before either species is predicted to go extinct.

b. If this were a biological control situation in which a parasite was introduced to

control a host species’ population, what single process-related parameter

would be most critical to keep the host population under control and/or to

drive it extinct? How might you go about determining what the “ideal”

parasite’s behaviour should be and whether or not that combination is

realistic?

c. Many species exhibit an Allee effect, which implies that they show negative

density dependence at both low and high abundance. The observation at high

abundance corresponds to what we see with logistic growth, and the

observation at low abundance could arise because populations at very low

density have a harder time encountering each other and hence do not have

sufficient numbers to reproduce and maintain a viable population. We can

include an Allee effect in our model for the prey population by changing the

equation to be

where A determines the abundance above which the prey population can

grow (and hence should be less than K). Add an Allee effect to the R code for

the model and determine how the parameter A impacts the dynamic

predictions of the model. You can try whichever hypothetical scenarios you

wish, but you might find repeating (a) to be a useful starting point.

d. If you wish to boost the prey population by introducing a third species, will it

be better to add an alternative prey for the predator or to add a species that

attacks the predator (e.g., a top predator or parasitoid)? Consider carefully

how many new parameters you need to add to describe the additional

biology/ecology, and try to start from the easiest case while working toward a

more “realistic” model. What assumptions do you need to make when adding

the new species?

5. Golden eagles, feral pigs, and insular carnivores

As discussed in the lectures, Roemer and colleagues made a number of model-based

predictions about the California Channel Islands that are a very good example of how

researchers employ mathematical models to understand global-change phenomena. The

primary article can be found at https://doi.org/10.1073/pnas.012422499 , and R code to

simulate and explore the author’s predictions can be found on Learn in the file roemer.R.