Dynamic behaviors of a commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure

Qifa Lin

doi:10.28919/cmbn/3652

Abstract

A two species commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure takes the form $$ \di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y }{d_1+y^2}\Big)-q_1Emx, \di\frac{dy}{dt}&=&y(a_2-b_2y)-q_2Emy $$ is proposed and studied, where $a_i, b_i, q_i, i=1,2$ $c_1$, $E$, $m(0<m<1)$ and $d_1$ are all positive constants. Depending on the range of the parameter $m$, the system may be collapse, or partial survival, or the two species could be coexist in a stable state. We also show that if the system admits a unique positive equilibrium, then it is globally asymptotically stable. By introducing the harvesting term and the reserve area, the system exhibit rich dynamic behaviors. Our results generalize the main results of Chen and Wu (A commensal symbiosis model with non-monotonic functional response, Commun. Math. Biol. Neurosci. 2017 (2017), Article ID 5).

Highlights

The aim of this paper is to investigate the dynamic behaviors of the following two species commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure dx dt c1y d1 + y2
Many scholars investigated the dynamic behaviors of the mutualism model ([1]-[12]), only recently did scholars pay attention to the commensal symbiosis mode, a model describes a relationship which is only favorable to the one side and have no influence to the other side([14]-[23]), commensal symbiosis is one of the relationship which could be observed in nature for many cases, for example: A squirrel in an oak tree gets a place to live and food for its survival, while the tree remains neither benefited nor harmed
Chen and Wu[13] proposed a two species commensal symbiosis model with non-monotonic functional response, they showed that the unique positive equilibrium of the system is globally asymptotically stable

Summary

Introduction

Sun and Wei[15] first time proposed a intraspecific commensal model: dx dt r1x k1 − x + ay k1 They investigated the local stability of all equilibrium points. Xie et al [22] proposed the following discrete commensal symbiosis model x1(k + 1) = x1(k) exp a1(k) − b1(k)x1(k) + c1(k)x2(k) , x2(k + 1) = x2(k) exp a2(k) − b2(k)x2(k) , They showed that the system (1.4) admits at least one positive ω-periodic solution. Wu[13] argued that this may not be suitable since the commensal between two species become infinity as the density of the species become infinity They proposed the following two species commensal symbiosis model with non-monotonic functional response dx dt.

The existence and local stability of the equilibria

Global stability of the equilibria

A COMMENSAL SYMBIOSIS MODEL WITH HARVESTING

Numeric simulations

Conclusion