# Difference between revisions of "Predator prey"

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| width=30% | [[Image:predprey-graph.png]] | | width=30% | [[Image:predprey-graph.png]] | ||

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==== Discrete Time Model ==== | ==== 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 <amsmath>H</amsmath> represent the population of hares and <amsmath>L</amsmath> represent the | ||

+ | population of lynxes, we can describe the state in terms of the | ||

+ | populations at discrete periods of time. Letting <amsmath>k</amsmath> be the | ||

+ | discrete-time index (e.g., the day or month number), we can write | ||

+ | {| width=100% | ||

+ | |- align=middle | ||

+ | | align=center | | ||

+ | <amsmath> | ||

+ | \aligned | ||

+ | H[k+1] &= H[k] + b_r(u) H[k] - a L[k] H[k], \\ | ||

+ | L[k+1] &= L[k] + c L[k] H[k] - d_f L[k], | ||

+ | \endaligned | ||

+ | </amsmath> | ||

+ | | (2.13) | ||

+ | |} | ||

+ | where <amsmath>b_r(u)</amsmath> is the hare birth rate per unit period and as a | ||

+ | function of the food supply <amsmath>u</amsmath>, <amsmath>d_f</amsmath> is the lynx mortality rate and <amsmath>a</amsmath> | ||

+ | and <amsmath>c</amsmath> are the interaction coefficients. | ||

+ | The interaction term <amsmath>a L[k] H[k]</amsmath> 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 <amsmath>c L[k] H[k]</amsmath> 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 <amsmath>x[0] = | ||

+ | (H_0, L_0)</amsmath> 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: | ||

+ | <center> | ||

+ | [[Image:predprey-discrete.png]] | ||

+ | </center> | ||

+ | Using the parameters <amsmath>a = c = 0.014</amsmath>, | ||

+ | <amsmath>b_r(u) = 0.6</amsmath> and <amsmath>d = 0.7</amsmath> 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. | ||

==== Continuous Time Model ==== | ==== Continuous Time Model ==== |

## Revision as of 16:10, 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).

#### 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

(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.