INTRODUCTION
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
them.
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.
|
