Difference between revisions of "Predator prey"
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Using the parameters <amsmath>a = c = 0.014</amsmath>, | Using the parameters <amsmath>a = c = 0.014</amsmath>, | ||
Revision as of 16:12, 21 October 2012
This page contains a description predator prey model that is used as a running example throughout the text. A detailed description of the dynamics of this system is presented in Chapter 3 - Examples and these dynamics are analyzed in Chapter 4 - Dynamic Behavior. A state space feedback controller is designed in Chapter 6 - State Feedback. This page brings together this material into a single place, to illustrate the application of analysis and design tools for this system. Links to MATLAB scripts are included that generate the analysis and figures described here.
System Description
The predator-prey problem refers to an ecological system in which we have two species, one of which feeds on the other. This type of system has been studied for decades and is known to exhibit interesting dynamics. The figure below shows a historical record taken over 90 years for a population of lynxes versus a population of hares (MacLulich, 1937).
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Figure 2.6: Predator versus prey. The photograph on the left shows a Canadian lynx and a snowshoe hare, the lynx’s primary prey. The graph on the right shows the populations of hares and lynxes between 1845 and 1935 in a section of the Canadian Rockies (MacLuluch, 1937). The data were collected on an annual basis over a period of 90 years. (Photograph copyright Tom and Pat Leeson.) | ||||
Discrete Time Model
A simple model for this situation can be constructed using a discrete-time
model by keeping track of the rate of births and deaths of each species.
Letting 
represent the population of hares and 
represent the
population of lynxes, we can describe the state in terms of the
populations at discrete periods of time. Letting 
be the
discrete-time index (e.g., the day or month number), we can write
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(2.13) |
where 
is the hare birth rate per unit period and as a
function of the food supply 
, 
is the lynx mortality rate and 
and 
are the interaction coefficients.
The interaction term 
models
the rate of predation, which is assumed to be proportional to the rate
at which predators and prey meet and is hence given by the product of
the population sizes. The interaction term 
in the
lynx dynamics has a similar form and represents the rate of growth of
the lynx population. This model makes many simplifying
assumptions -- such as the fact that hares decrease in number only
through predation by lynxes -- but it often is sufficient to answer
basic questions about the system.
To illustrate the use of this system, we can compute the number of lynxes
and hares at each time point from some initial population. This is
done by starting with 
and then using equation (2.13) to compute
the populations in the following period. By iterating this procedure, we can
generate the population over time. The output of this process for a specific
choice of parameters and initial conditions is shown below:
Using the parameters 
,
and 
in
equation~\eqref{eq:modeling:predprey} with daily updates, the
period and magnitude of the lynx and hare population cycles
approximately match the data in Figure 2.4. While the details of
the simulation are different from the experimental data (to be
expected given the simplicity of our assumptions), we see
qualitatively similar trends and hence we can use the model to
help explore the dynamics of the system.
