As human kind moves into the twenty-first century and our global population continues to grow at an astonishing rate, it will become increasingly important to be able to use mathematical models to observe nature’s populations.  Through careful use of predator-prey models, human kind can monitor animal populations to help insure a minimal negative impact on the environment.  There have been many predator-prey models created for different purposes over the years, such as the Lotka-Volterra model (below).  The main purpose is to predict and understand the trends that are related to the species involved.  Prediction can aid in resource management and planning for future generations. 
    To construct the type of predator-prey model that will be presented here,
 start with two variables: one for the prey and another for the predator. The
 predator and prey each must have an associated growth rate. Whenever an
 interaction occurs between the two, there are two scenarios that can take
 place: the predator can kill and consume the prey or the prey can escape. 
 For these two options there is a probability variable for each.  The model
 includes the rate at which the predators are likely to make offspring of the kill,
 and the rate at which the prey decrease due to the predators consuming

     The predator-prey models only work in settings where there are no other animals involved in this competition. In real life, it is common for there to be more than one type of predator for the prey and/or more than one type of prey for the predator. This model does not account for that. This is an interaction between only two species. Also, it must take place in a closed system; in other words, there must be no immigration or emigration. The only growth to the population are the newborns and a decay in population must result from death.  The following is an example of one of the most common types of predator-prey model equations (the Lotka-Volterra model) :

Pn =  (1 + a) (Pn-1) – b (Pn-1) (Qn-1)    

Qn =  (1 – d) (Qn-1) + c (Pn-1) (Qn-1)    

    Q = predator population.
    P = prey population.
    a = the natural rate of growth of the prey population if there are
          no predator.
    b = the rate of decrease of prey due to encounter with predators.
    c = the rate of predator increase due to interactions with the prey.
    d = the natural rate of decay of the predator population if there
          are no prey.
    n = the current generation.
      In the model, if the prey is kept in isolation, they will reproduce and increase at an exponential rate.  The exponential growth model describes this situation, Pn = (1 + a) (Pn-1), where Pn is the prey population for the nth  generation.  Factoring the equation produces Pn = Pn-1 + (a * Pn-1); where the previous population is Pn-1 and the rate that they grow times the population count is the number of newborn.  Under exponential growth, the prey will grow without bound. 
      On the contrary, without the presence of prey the predators  will lack the food necessary to survive and become extinct. The exponential decay model, Qn = (1 - d) (Qn-1), describes the predators population.  It is exactly the same as the prey population except the rate is negative. The predators cannot grow without the prey because they lose their only source of food. 
     When interaction occurs, the predator population benefit and the prey population are jeopardized. In the Lotka-Volterra model, the term (Pn-1) (Qn-1) reflects the possible predator-prey interaction during the previous generation. Of all the possibilities mentioned, only one can happen at each time. It is also likely that more than one prey may be necessary for the predator. These probabilities combined lead us to our final equations presented above.  The graphs on the following pages display different views of the Lotka-Volterra models.

Intro    Pred-Prey Graphs    Phase Plots    Equilibrium Points    Conclusion&Error

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