Predator Prey Model Using Differential Equations

The predator prey model is characterized by the predator and prey. An example of this can be a parasite and its host. The model shows that both predator and prey depend on each other. For instance, in the absence of the predator, the prey grows while if the prey is absent, the predator starves. Therefore, the prey and predators influence the evolution of each other. To apply differential equations in the relationships between the predator and prey, one has to consider some variables.

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For instance, P=predators, N=prey, t=time, r=prey’s rate of growth, a=rate of attack, q=mortality rate of predator and c=the efficiency of the predator to turn food to offspring. If the prey is absent, the numbers of predators will exponentially decline. This is shown by: dP/dt = -qP. The negative shows that there will be a decrease in the predator numbers. On the other hand, if the prey is present, there will be increased numbers of predators.

However, this will be determined by the rate of consumption given by acPN. As the number of predators(P) and prey(N) increase, there are more frequent encounters. However, the consumption rate is dependent on the rate of attack,a. Thus, the equation representing the dynamics of the predator will be: dP/dt = caPN-qP. In the absence of the predators, the preys are expected to increase exponentially.

This can be represented by: dN/dt = rN. If predators are present, the prey would not have an exponential increase. Therefore, the prey’s dynamics will be represented by: dN/dt = rN – aPN. The predator prey model is made of the predator and prey dynamics and thus gives a prediction of the relationship that exists between the predator and prey (in the presence of each other).

Reference

Lotka-Volterra, Predator-prey dynamics. Retrieved on March 2, 2011 from: http://www.tiem.utk.edu/bioed/bealsmodules/predator-prey.html.

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