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

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.