Dynamic Behavior of a Commensalism Model with Nonmonotonic Functional Response and Density-Dependent Birth Rates

In this paper, we propose and analyze a commensalismmodel with nonmonotonic functional response anddensity-dependent birth rates. The model can have at most four nonnegative equilibria. By applying the differential inequality theory, we show that each equilibrium can be globally attractive under suitable conditions. However, commensalism can be established only when resources for both species are large enough.


Introduction
Commensalism is a long-term biological interaction in which members of one species gain benefits while those of the other species neither benefit nor are harmed.An example of it is that remora are specially adapted to attach themselves to larger fish that provide locomotion and food.In the last decades, commensalism has attracted the attention of many researchers ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]).Complicated dynamics have been found in the study.For example, in [3], Lin considered the effects of partial closure and harvesting.Depending on the size of the harvesting area, species can go extinct, partially survive, or become permanent.He also showed in [4] that the final density of the species increases as the Allee effect increases.This is quite different from results for predator-prey system with Allee effect.
When the interaction between the species is ignored, the growth for both species is described by traditional logistic equations.Indeed, without the presence of , the growth of the first species takes the form where  1 is the intrinsic growth rate and  1 is the densitydependent coefficient or the interspecific competition coefficient.However, in most situations, the intrinsic growth rate For more details, see [6][7][8].Combining this with (1), we propose the following commensalism model: where   ( = 1, 2,  = 1, 2, 3, 4) and  1 ,  The aim of this paper is to investigate the attractivity of equilibria of (5).The main tool is the differential inequality theory or comparison principle.To the best of our knowledge, this is the first time to use differential inequality in this direction for ecosystems.The rest of the paper is arranged as follows.In Section 2, we obtain the existence and global attractivity of equilibria of system (5).Section 3 is devoted to illustrating the feasibility of the main results through numeric simulations.We end this paper by a brief discussion.

Numeric Simulations
In this section, we provide numeric simulations to illustrate the four situations in Theorem 3.

Discussion
In this paper, inspired by the work in [17][18][19], we proposed a commensalism model under the assumption that the intrinsic growth rates of both species are density-dependent.
The model can have at most four equilibria.For the first time, differential inequality has been applied to obtain the global attractivity of equilibria of such ecosystem models.Depending on the availability of resources, each of the possible equilibria can be globally attractive.This implies that density-dependent birth rates play an important role in the dynamics.Though the dynamics can be complicated, from the point view of commensalism, commensalism can be established only when resources for both species are large enough (see Theorem 3 (iv)).Hence, these results agree with those of Chen and Wu [5] (see Theorem A in Introduction).As we know, delay always exists in many biological processes.We will leave the effect of delay on the dynamics for future study.
1,  1 , and  2 are all positive constants.Here () and () are the densities of the first and second species at time , respectively. 11 and  21 stand for the total resources available per-unit-time for species  and , respectively.