Dynamic Analysis and Optimal Control of a Fractional Order Singular Leslie-Gower Prey-Predator Model
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A FRACTIONAL ORDER SINGULAR LESLIE-GOWER PREY-PREDATOR MODEL∗
ê')
Linjie MA (
4R)
Bin LIU (
†
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China E-mail : [email protected]; [email protected] Abstract In this article, we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model, which describes the interaction between populations of prey and predator, and takes into account the economic interest. We firstly obtain the solvability condition and the stability of the model system, and discuss the singularity induced bifurcation phenomenon. Next, we introduce a state feedback controller to eliminate the singularity induced bifurcation phenomenon, and discuss the optimal control problems. Finally, numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior. Key words
fractional order system; differential-algebraic system; prey-predator bioeconomic model; singularity induced bifurcation; optimal control
2010 MR Subject Classification
1
34A08; 34A09; 92B05; 37G10; 93D15
Introduction
The prey-predator is a popular and important prototype model, which has attracted a great deal of attention due to its universal existence and induced diverse applications. In particular, it plays a crucial role in the management of renewable resources in bioeconomic systems (see for example, [1–3]). One prey-predator model is the modified Leslie-Gower model with MichaelisMenten type prey-harvesting dx x a1 xy qEx dt = rx(1 − d ) − n + x − m E + m x , 1 1 2 (1.1) dy a2 y = sy(1 − ). dt n2 + x
Here x(t) and y(t) are the population sizes of prey and predator, respectively; E is the harvest; the parameters r and s are intrinsic growth rates of prey and predator, respectively; d is the ∗ Received
September 17, 2018; revised September 30, 2019. This work was partially supported by NNSFC (11971185). † Corresponding author: Bin LIU.
1526
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
environmental carrying capacity for the prey; a1 and a2 are the maximum values of the per capita reduction rate of prey and predator; n1 and n2 measure the environmental protection to prey and predator, respectively; q is the harvest ability coefficient; m1 and m2 are constants. If all the parameters are positive, we say that the model is biologically feasible. The Leslie-Gower model (1.1) has been studied by many authors. For dynamic behavior, the global stability and existence of periodic solutions of model (1.1) were investigated in [4]. [5] and [6] discussed the global dynamics of model (1.1) with stochastic perturbation. [7] mainly investigated local bifurcations of model (1.1) consisting of sadd
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