Feedback Control and Parameter Invasion for a Discrete Competitive Lotka–Volterra System

Li Xu,Shanshan Lou,Guang Zhang,Panqi Xu

doi:10.1155/2018/7473208

Li Xu, Shanshan Lou + Show 2 more

Open Access

https://doi.org/10.1155/2018/7473208

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Abstract

State feedback is used to stabilize the Turing instability at the unstable equilibrium point of a discrete competitive Lotka–Volterra system. In addition, a regularization method is applied to parameter inversion for the given Turing system and numerical simulation can verify the effectiveness of the algorithm. Furthermore, how less or more sample data and dependence on the initial state affect estimation procedure are tested.

Highlights

In theoretical ecology, the models governed by difference equations are used to characterize the interactions of species when the size of the population is rarely small [1]
In order to stabilize the orbit at an unstable equilibrium point of system (2), we use the state feedback control method and indirect control variables are added; we can get the system xti,+j1 = rxti,j (1 − xti,j − ayit,j − b1uti,j) + D∇2xti,j yit,+j1 = ryit,j (1 − axti,j − yit,j − b2Vti,j) + D∇2yit,j (10)
{0 other, comparison experiments are done using the regularization method (RM) and least square method (LS); Table 2: The estimation result when the sample values are generated from special initial value from t = 1 to t = 10000

Summary

Introduction

The models governed by difference equations are used to characterize the interactions of species when the size of the population is rarely small [1]. The partial difference systems (space-time discrete systems) for biological patterns resulted from diffusion-driven instability (Turing instability) in plants and animals and the set of equilibrium patterned solutions have been studied in great detail over the last several years; for example, see [6,7,8,9,10,11] To the best of the authors’ knowledge, there are still no scholars who are investigating the stability property of the 2D spatially discrete reactiondiffusion competitive system with feedback controls; this motivates us to propose such a model as follows: xti,+j1 = r1xti,j (1 − a11xti,j − a12yit,j − b1uti,j) + D∇2xti,j yit,+j1 = r2yit,j (1 − a21xti,j − a22yit,j − b2Vti,j) + D∇2yit,j (5).

Feedback Control and Its Stability

Parameter Inversion

Conclusion