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The Lotka-Volterra predator-prey model is widely used in many disciplines such as ecology and economics. The model consists of a pair of first-order nonlinear differential equations. In this paper, we first analyze the dynamics, equilibria and steady state oscillation contours of the differential equations and study in particular a well-known problem of a high risk that the prey and/or predator may end up with extinction. We then introduce exogenous control to reduce the risk of extinction. We propose two control schemes. The first scheme, referred as convergence guaranteed scheme, achieves very fine granular control of the prey and predator populations, in terms of the final state and convergence dynamics, at the cost of sophisticated implementation. The second scheme, referred as on-off scheme, is very easy to implement and drive the populations to steady state oscillation that is far from the risk of extinction. Finally we investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the convergence guaranteed scheme achieves theoretically optimal performance, the on-off scheme is more attractive for practical applications.

Predator-prey population dynamics are often modeled with a set of nonlinear differential equations. The Lotka- Volterra model [

where

Without the predator

Without the prey

The above Lotka-Volterra model (1) describes the autonomous dynamics between the two species without an exogenous input. In this paper, we introduce exogenous control to change the dynamics so as to achieve certain desired characteristics. In particular, it is well known [

We plot the predator and prey populations in the

To see this, note that

It follows that

Plot of predator and prey populations varying with time

Plot of predator versus prey population contours. a = d = 0.5, b = c = 0.01. “+” represents one of the two equilibrium points

There are two solutions:

An important observation from both

While the particular dynamic pattern may appear to be an artifact of the Lotka-Volterra model, the above ob- servation demonstrates certain phenomenon in some real world ecological environment where the predator-prey dynamics are so out-of-balance that the prey and/or predator end up extinction. In the following we will intro- duce a dynamic control mechanism to the Lotka-Volterra model so as to reduce the risk of extinction.

Exogenous control mechanisms have been studied in the literature [

With exogenous control, the dynamic system is in general described as follows

where

population to a desired final state

ward the final state is also to be controlled. As an example, suppose that the control scheme is required to drive

from any initial population

ential equation

Constant

Constants

The control scheme is thus given by

It follows that

Because the population is guaranteed to converge to the final state, this control scheme is called convergence guaranteed scheme. The control scheme is expressed in a closed-loop feedback form. From the closed form ex- pressions of

Clearly the control is stable and converges to the steady state

gence rate depends on

Note that in the convergence guaranteed control scheme the control variables

The above convergence guaranteed control scheme (7) achieves very precise control of the population—not only the final state but also the trajectory to the final state is precisely specified. In many real world scenarios, such a high precision may not be required. In the following, our objective is to control the population to be within a de- sired dynamic range that is far from the risk of extinction. By relaxing the control objective, we aim to greatly simplify the implementation of the control scheme.

Specifically, consider the following on-off control scheme

Here,

The key of the on-off control scheme is to design the control condition

where

One way to view this on-off control scheme is to plot the predator versus prey population. Recall that with no control, the predator and prey populations of the original autonomous dynamic system (1) oscillate along a con- tour. If the control is constantly exerted, i.e.,

The second choice is based on the first-order derivative of the population of the prey

where

We examine the performance in

Performance of the convergence guaranteed control scheme. a = d = 0.5, b = c = 0.01. x(0) = y(0) = 10. x_{T} = 100, y_{T} = 20

Control u_{x}, u_{y} in the scenario of Figure 3

Performance of the on-off control scheme with con- trol condition (11) and x_{0 }= 100, g = 1

Contour plot of Figure 5

Performance of the on-off control scheme with con- trol condition (12) and g = 2

pattern, control is only exerted very occasionally. We make a similar observation for a different initial condition, e.g.,

In this section, we investigate the robustness of these two schemes against parameter mismatch. Specifically, so far we have assumed that the control scheme knows exactly the model parameters

To study the robustness, we assume that the actual parameters are slightly different from those used in the control scheme. That is, the control scheme assumes the following set of parameters:

Plot of predator versus prey population of the case of α = −20, x(0) = y(0) = 10 in Figure 7

Robustness against parameter mismatch

The parameters of the actual model are given by

In this paper, we have proposed two control schemes to reduce the risk of extinction in the Lotka-Volterra pre- dator-prey model. The convergence guaranteed scheme achieves very fine granular control of the prey and pre- dator populations, in terms of the final state and convergence dynamics, at the cost of sophisticated implementa- tion. The on-off scheme is very easy to implement and drive the populations to steady state oscillation that is far from the risk of extinction. We furthermore investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the conver- gence guaranteed scheme achieves theoretically optimal performance, the on-off scheme is more attractive for practical applications. A next step of the research would be to apply the on-off scheme to other predator-prey models and investigate the robustness not only against parameter mismatch but also against model mismatch.