# Complex Dynamical Behavior in Bio-economic Prey-Predator Models with Competition for Prey

• Qingling Zhang
• Chao Liu
• Xue Zhang
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 421)

## Introduction

In recent years, harvesting in the prey-predator ecosystem is one of the most important fields of interest. Much research effort [1, 3, 2, 14, 11, 16, 5, 6, 8] has been put into investigating the interaction and coexistence mechanism of the harvested prey-predator ecosystem. A numerical analysis of a harvested prey-predator model is performed in [1], and the model can be expressed as follows:

$$\left\{ \begin{array} {l} \dot{x}_1(t)=x_1(t)(\epsilon_1-a_{11}x_1(t)-a_{12}x_2(t)-a_{13}y(t)), \\ \dot{x}_2(t)=x_2(t)(\epsilon_2-a_{21}x_1(t)-a_{22}x_2(t)-a_{23}y(t)), \\ \dot{y}(t)=y(t)(-\epsilon_3+a_{31}x_1(t)+a_{32}x_2(t))-H(t), \end{array} \right. (13.1)$$

where x 1(t) and x 2(t) denote populations of the prey 1 and prey 2, respectively; y(t) represents population of the predator; ε i (i = 1,2,3) are intrinsic rates of growth and decay of the three species; a ij (i,j = 1,2,3) with i ≠ j are the inter-species and a ii (i = 1,2) are the intra-species coefficients of competitive interactions; a 13 and a 23 are the coefficients for the loss of prey 1 and prey 2, respectively; a 31 and a 32 are the coefficients for growth in the predator as a result of consumption of prey by them; and H(t) is a harvest function, which represents harvesting strategies applied to the predator.

## Keywords

Economic Interest Feedback Gain State Feedback Controller Interior Equilibrium State Feedback Control
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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