Now that you thoroughly understand population regulation (see here, here and here), let’s start developing some more sophisticated models where interactions with features of the environment – namely other species – regulate the abundance of species. We’ll start this exploration by considering a very simple model of a predator feeding on a single prey species. This model was first proposed independently by Alfred Lotka in 1925 and Vito Volterra in 1926. To specify a model, one must first state what assumptions will be used to construct the model. We’ll develop this model by explictly stating the assumptions and then turning these assumptions into mathematics. Assumption 1: Two species are being modeled – a predator species and a prey species. The abundance of the predator will be denoted P, and the abundance of the prey will be denoted N. Assumption 2: The abundances of the two species will respond to one another instantaneously, and so their abundances will be modeled in continuous time. Therefore, we will use differential equations to represent the changes in their abundances with time: and . Assumption 3: In the absence of the predator, the prey species follows exponential population growth. Thus, the basic model of prey population growth is:
Remember that r describes the balance between per capita birth and death rates. However, here r includes all sources of death that do not include the predator. The per capita death rate imposed by the predator will be modeled separately. Assumption 4: The predator causes an additional set of deaths in the prey, based on the frequency of encounters between the predator and prey. We will assume the most simple model of encounters here, called mass action. Mass Action assumes that encounters between the predator and prey individuals occur at random, just as though they were balls bumping into one another as they roll around on a table. Thus the encounter rate between predators and prey is simply the product of their abundances: N times P.
We will further assume that only a fraction of these encounters result in a dead prey that is eaten by the predator. Thus, the total death rate due to predation is aNP, where a is the parameter that scales encounters into a death rate. a is called the attack coefficient. This is known a Type I functional response.
The full model for the change in the prey population over time then has an additional term due to predator deaths. The full prey equation is
The first term (rN) describes exponential population growth in the absence of the predator, and the second term (-aNP) is the death rate due to the predator. Assumption 5: Now we turn to the predator. The predator is assumed to be a specialist that only feed on this prey. Furthermore, we will assume that new predator babies produced in direct proportion to the number of prey that each predator kills. The parameter b will be used to represent the conversion efficiency of the predator, namely the number of predator babies produced for each prey eaten pe unit time. Thus, the rate at which the predator population produces babies is simply the product of how many prey are killed per unit time (aNP) times the number of babies produced for each prey killed (b), which is baNP. This term is also known as the numerical response of the predator to prey abundance. Assumption 6: The predators are also assumed to die at a constant per capita rate, given by the parameter d. The total death rate of the predator is then dP, the per capita death rate times the number of predators. Given assumptions 5 & 6, we can now write down the full equation for the change in the predator population with time, which is simply birth rate minus death rate.
Thus, the full model is a pair of coupled differential equations that describe the effects of predator and prey on one another’s abundances.
The coupling is based on how the various demographic rates depend on the abundances of the two species – note the N‘s and P‘s in each equation. This is what couples their population dynamics.
Remember that population regulation requires density dependence, and density dependence must be evaluated based on per capita population growth rates. So to understand density dependence in this model, rewrite these two equations as per capita population growth equations. In other words, divide the prey equation by N and divide the predator equation by P.
When written this way, we see that the per capita growth rate of neither species depends on its own abundance, but rather only on the abundance of the other species. The per capita growth rate of the prey only depends on the abundance of the predator (i.e., the prey per capita equation only has P in it but no N), and the per capita growth rate of the predator only depends on the abundance of the prey (i.e., the predator per capita equation only has N in it and no P). This will have important implications for the stability of the equilibria in this model, but we’ll get to that shortly. Remember that we follow a standard procedure in analyzing such population models. First, we must find the equilibria of the model, and then evaluate the stablity of these equilibria. As with the analyses of single population models, the equilibria are found by setting the total population growth rate equations equal to zero, and determining the abundances of species where they do not change. However, now we have two equations to deal with, so we set both equal to zero and simplify.
In this model, this procedure results in two very simple equations. These equations are called zero net growth isoclines because they map out the abundances at which a species has a zero net growth or simply isoclines. The isocline functions will be very important in understanding the dynamics and stability of the model. The prey isocline is given by the function P=r/a, and the predator isocline is given by the function N=d/(ab). Each function maps out the abundance of that species when it will not change in abundance. In other words, all points along each are equilibria for that species. All points where these two functions intersect are joint equilibria where both species are at equilibrium, and so neither will change in abundance. Ultimately, we are searching for these joint equilibria, and so we will refer to these as the equilibria of the model.
An equilibrium of the model is a combination of abundances of the two species where if they start jointly at these abundances, they will remain there. In other words, the abundance of neither species will change when the system is at one of these joint abundances that is an equilibrium.
This applet runs a model of the basic Lotka-Volterra predator-prey model in which the predator has a Type I functional response and the prey have exponential growth. The red line is the prey isocline, and the red line is the predator isocline. Move the sliders to change the parameters of the model to see how the isocline positions change with changes in parameter values. Click on any point in the figure to see how predator and prey abundances change with time. The trajectory may go out of the frame for some starting points. (Note that this is a Java applet, and you may need to change your security settings in the Java Control Panel on your computer so that it will run.)
[applet code="LVExponentialTypeIApplet" file="http://www.enallagma.com/wordpress/wp-content/uploads/2012/05/LVExponentialTypeIApplet.jar" width="420" height="650"]