The first model we will look at is the logistic equation. This was presented in the lecture as an underlying framework for thought regarding population dynamics.
The classic equation is written as:
\(\frac{dn}{dt} =rN(1-\frac{N}{K})\)
\(\frac{dn}{dt}\) is the notation used to represent the rate of change of the population.
There are two parameters in the model.
r is the intrinsic rate of increase (or decrease) of the population. As the first term is written as \(rN\) with N being the population size this term produces exponential growth.
The second parameter is K. This represents the carrying capacity. When the population reaches the carrying capacity \((1-\frac{N}{K}) = 0\). So \(rN(1-\frac{N}{K}) = 0\) and there is no population growth at all.
The model can also be run with discrete change (this is a rather technical idea that those who have not studied calculus would not be aware of ). The discrete model is written as
\(\frac{\delta n}{\delta t} =rN(1-\frac{N}{K})\)
Or as
\(\frac{\Delta n}{\Delta t} =rN(1-\frac{N}{K})\)
In this case it is possible for the population to “overshoot” the carrying capacity. In this case \((1-\frac{N}{K}) < 0\), i.e. it becomes negative, and the population falls back.
The equation can be explored using two runs in order to compare the resulte of changing the parameter.
http://r.bournemouth.ac.uk:3838/Ecosystems/Logistic/
What happens to the initial slope of the curve as r is increased?
What happens to the shape of the curve is K is changed?
At what value for r does an “overshoot” of the carrying capacity occur?
If the overshoot is small what happens next?
As the overshoot becomes larger what happens to the figure? How can this be described in words?
When the overshoot is very large what happens to the subsequent dynamics?
How does the initial starting value for the population affect the result over time?
What is this effect known as?
How does it differ from a random (stochastic) effect?
Formally the mathematical representation of a leslie matrix looks like this.
\(\begin{bmatrix} f_0 & f_1 & f_2 & f_3 & f_{4+}\\ s_0 & 0 & 0 & 0 & 0\\ 0 & s_1 & 0 & 0 & 0\\ 0 & 0 & s_2 & 0 & 0\\ 0 & 0 & 0 & s_3 & 0 \\ 0 & 0 & 0 & 0 & s_{4+} \end{bmatrix} \begin{bmatrix} n_0 \\ n_1 \\ n_2 \\ n_3 \\ n_{4+} \end{bmatrix}\)
The terms labelled \(f_0 f_1\) etc are fecundities.
The terms labelled \(s_0 s_1\) etc are survival probabilities (note that mortality probability = 1-survival probability)
The terms labelled \(n_0 n_1\) etc are the numbers of individuals in each age class.
Without going into more details regarding matrix algebra, the model can be “run” by multiplying the matrix by the vector containing the age classes.
When this is done a rather interesting effect emerges. It doesn’t matter what vector is initially chosen for the age classes. After iterating the calculation the result eventually settles on a stable age structure. Another interesting result is that all Leslie matrices also settle down to produce either exponential growth or decay of the population. The stable age structure is known as an eigenvector and the rate of growth (or decay) is called the eigenvalue. This implies that the model does not consider any carrying capacity. So it cannot predict population dynamics over long time frames, as the values themselves will change as the population changes. However the model can be very useful for short term projections. Human demographers often have good data to set up both the age class vector and the leslie matrix. So this is a useful tool for projecting the growth of a population within a country.
Ecologists on the other hand very rarely have good data to parametrise the model, although there are some published papers which have managed to use it. Nevertheless the model can be useful for obtaining an estimate of the stable age structure implied by estimates of mortality and fecundity in wild populations.
The link below allows you to easily set up a Leslie matrix for a population consisting of 4 age classes. This is reasonable for small mammals and small birds. The same model can be used for longer lived organisms by assuming that the classes are rather wider (2 or more years rather than 1).
http://r.bournemouth.ac.uk:3838/Ecosystems/leslie/
The matrix is “solved” which produces the eigenvalue and the stable age class distribution. Estimating the mean number of young produced each year and the probability of surviving through a year can be simpler than trying to measure the age structure. So the model can be used as a “thought tool” to consider how populations might be structured. It also provides a rough idea regarding how fast a population might increase.
Which age class usually contains the highest proportion of the population?
Try “guestimating” the parameters for a population of blue tits or voles. What data would you need to make these estimates more reliable?
How does lowering the survival of this age class affect the eigenvalue?
How might you design a management strategy to reduce this population?
What might affect fecundity in a natural ecosystem?
What might affect survival in a natural ecosystem?
What affect would predation of the oldest member of the population have on the absolute numbers in the population?
The mathematicians Alfred Lotka and Vito Volterra are most famous for proposing a model that represents the dynamics of a predator-prey system. However they also extended the logistic equation to represent competition. They did this by proposing a shared carrying capacity that all species competed for.
The model below assumes that different species have different intrinsic growth rates and different carrying capacities, whilst all contributing to the consumption of a shared resource.
When run without disturbance the K selected species eventually “wins out” through successfully competing for all the shared resources. But what happens if there are periodic disturbances?
What sort of disturbance regime leads to long term coexistence of the highest number of species?
What sort of species benefit from frequent disturbance?
How might this conceptualisation be useful when proposing management strategies?
Lotka and Volterra are more famous for the model representing predator and prey dynamics. The model is in effect extremely simple. It assumes “top down” control of prey numbers through predation. It also assumes only one type of predator and one sort of prey. All prey mortality is due to predation and predators gain energy (and thus are able to reproduce) through consuming prey. In effect there is a conversion factor between prey and predators. For example a fox might requires ten rabbits to gain enough energy to give birth to another fox.
The model has been implemented here as an insight maker model.
https://insightmaker.com/insight/4DYFdNOfy5alJd395LWc12/Lotka-volterra
The insight maker shows the structure of the model in the compartment-flow convention. It can be run by clicking on simulate.
How would you describe the dynamics predicted by the model?
Which of the two populations reaches its peak first?
Why does the prey population decline after reaching a peak?
Why does the predator population decline after reaching a peak?
How “realistic” is this model?
List all the factors you can think of that the model does not include.
One simple modification to the Lotka Volterra model involves giving the prey population a carrying capacity.
https://insightmaker.com/insight/75OLTTaXUZ8qNJKwBHJ43d/Lotka-volterra-with-carrrying-capacity
This is a very simple “toy” individual based model
http://r.bournemouth.ac.uk:82/Ecosystems/Wolf%20Sheep%20Predation.html