The dynamics of two populations, one predator the other prey, are modeled using the Lotka-Volterra equations. In this post I’ll discuss why these equations have the form they do in a fully mixed model, and look at generalizations to incorporate spatial dynamics.

Let’s start with basic growth. If you have a large population then it will have more children than a small population. The rate of growth of the population is proportional to its size. Denoting the number of individuals in a population i at time t as N_i (t) we can write

https://i2.wp.com/mathurl.com/hsvo494.png

Here the growth rates are functions of the other populations. We want this because the size of a predator population should affect the death rate of a prey population, and the size of the prey population should affect the birth rate of a predator population. For a two population model we have then:

https://i2.wp.com/mathurl.com/jzzflrv.png

https://i1.wp.com/mathurl.com/h5djb6e.png

For a prey population the alpha is positive (base birth rate higher than death rate) and the beta is negative (death rate increased by presence of predator), and for a predator population we have the reverse (base death rate higher than birth rate, birth rate increased by presence of prey). This basic system has oscillatory dynamics: the predator population stays low until there is a surge in prey population. The surge in the predator population forces down the prey population, which in turn leads to a collapse of the predator population allowing the prey to grow again.

Now how do we add spatial dynamics?

First we allow the population functions to be functions of space, and define population densities so that the number of individuals in a region \Delta \vec r around some point \vec r is N_i(\vec r, t|\Delta \vec r)=n_i(\vec r,t)\Delta\vec r. What kind of dynamics do we wish to incorporate? How about the exploration of the environment, as well as the propensity for predators to chase prey and prey to run away from predators?

https://i0.wp.com/mathurl.com/htetwvt.png

https://i0.wp.com/mathurl.com/z3wc25d.png

With no environmental pressure we might think of the individuals in a population as performing random walks. High population densities will diffuse to regions of lower density, and in an idealized limit there would be individuals everywhere in the environment. Overpopulated lumps dissipate and underpopulated dips fill up. By noting that lumps have negative curvature while dips have positive curvature, this dynamic can be implemented with a Laplacian:

https://i2.wp.com/mathurl.com/zza29m6.png

If this was the only term on the RHS, then we would have the diffusion equation, where the motility, \mu, of the species is playing the role of the diffusivity. It is a positive number, and the larger it is the faster the individuals in the species can move around. Clumping behavior could be modeled by allowing the motility to become negative.

Alright, so what about the coupled spatial dynamics of the two populations: chasing and escaping? What kind of term can model this? Consider the density of both populations at a single point and let’s think about how one of the populations is reacting to the other. If the population is predatory then it will want to move in the direction of increasing prey population. A prey will act in the opposite fashion. In an interval of time the individuals at a point will shift to a new point that is determined by the gradient of the other population, hence:

https://i1.wp.com/mathurl.com/h6zejvc.png

Chasing and escaping behavior can be modeled with gradient couplings! Here \nu is the coupling motility, and is positive for chasers (predators) and negative for escapees (prey).  Our spatial dynamics are captured in the operator:

https://i2.wp.com/mathurl.com/jdut7b8.png

In the case of two populations we now have:

https://i0.wp.com/mathurl.com/grsy9fl.png

These equations are a great starting place to start looking at the spatial dynamics of populations, and it is pretty obvious about how one can go about and generalize them to to include more complicated behavior.

Just to give you a sense of how the spatial predator and prey dynamics evolve over time I coded a slightly more complicated model than the one just written (included a predator-predator fighting component):

Also, make sure to check out the artwork of Alex Solis if you dig adorable versions of predators and prey.

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