Dynamics of a two-species model with transitions in population interactions

A two-species model with variable population interactions and harvesting on one of the species is studied. Existence and stability of equilibria and existence of periodic solutions are established, existence of some bifurcation phenomena are analytically and numerically studied, explicit threshold values are computed to determine the kind of interaction ( mutualism, competition, host-parasite) between the two species. A brief discussion on the influence of the harvesting function on the dynamics of the model is also included.


Introduction
The population models with static interactions, like competition, predator-prey, mutualism, is widely and well studied in [2], [6], [7], [11], [14]. However, there are some evidences [5], [12], [13], [14] show that it not adequate the case as the interspecific relationship may change depending on population densities and environmental parameters and so on. Several studies [5], [4], [15] have focus on condition interaction which is represented via nonconstant functions. Jorge Rebaza [10] studied the model incorporates linear α-functions representing the variable interaction between the species, and also considered harvesting on one of species by an external agent. For the first time, Hopf bifurcations and periodic solutions are found in this kind of models. The aim of this paper is to consider more general and rational α-functions representing the variable interaction between the species, and discuss the main influence of harvesting on the dynamics of the model.
In the present paper, we consider the following model: are the population densities of species 1 and 2 respectively, and all the parameters are positive. The constants 1 r , 2 r are the corresponding intrinsic is the harvesting function which is increasing smoothly with the size u of the population, h is the rate-of-harvesting limit, and e is the number of species 1 it take to reach one half of the maximum harvesting rate. We consider the functions 1  , 2  represent the interactions between both species, where changes in environmental conditions. So 1  , 2  can take positive or negative values, and thus the interspecific interactions are not fixed but vary with the environmental parameters and the system state, and can take positive or negative values. Classically, one considers 1  , 2  as fixed parameters, and their signs determine the interspecific interaction. In particular, if 1 , then the model reduces to the one studied in [1], [10].
The paper is organized as follows. In section 2, we studied the existence and local stability of the equilibrium. The existence of certain bifurcations both analytically and numerically is resolved in section 3. In section 4, the impact of harvesting on one of the species is discussed.

Local stability of equilibria
The system (1), (2) determines the following equilibria: Thus  u is real if   From the second nullcline in (4), we get v is a convex curve about u . So we can establish conditions under which there is at least one solution of (4), with 3 u > 0,   (4), we impose a negative slope on the first nullcline at the v -intercept, that is The general Jacobian 0 A of (1): In the theorem below, we denote Theorem 2.1. The boundary equilibrium points of (1) have the following local stability properties: is a saddle or a stable/unstable node according to 1. if P is a saddle, and a stable node if the inequality is reversed.
then 1 P is a stable node, and a saddle if all the inequalities but the first one are reversed. 3. if e r h 1  ,  then 1 P is a saddle, and an unstable node if P is never a focus nor of center type.
in (3), we see that if if all the inequalities are reversed but the first one. Combining the inequalities above, the first three statements follow. In addition, 1 P is never a focus nor of center type, as none of the eigenvalues is complex.
.Clearly, 2 P is never a focus nor of center type.

Interspecific relationships
For the coexistence equilibrium there is mutualism at 3 P when 1 D > 0 and 2 D > 0, that is when ; there is competition at 3 P when   u u and   v v and host-parasite at 3 P when either   u u and Host-parasite when ). Remark. These inequalities provide with threshold value that determine the kind of relationship between the species.
In the theorem below, we denote , v u P  of (1) has the following stability properties: (a) It is locally asymptotically stable if 0  B and and unstable if the last inequality is reversed. (e). It is of center-type if Proof. Using the nullcline equations (4), Jacobian at 3 P can be written as Thus, the eigenvalues are Through analysis symbols of 1  and 2  , we confirmed the above conclusions.

Bifurcations
Equations should be centred and should be numbered with the number on the right-hand side.
,Thus , using Sotomayor's theorem one concludes that under the condition above, the system (1) undergoes a transcritical bifurcation at 0 P . Similarly, at 2 P : Hence, under the given condition, there is a transcritical bifurcation at 2 P . Theorem 3.2. The system (1) undergoes a saddle-node bifurcation at 1.
, if any of these cases hold: (a).
Proof. 1. The general Jacobian at 1 P reduces to   .
where ij a and ij b are the coefficients of the expansions.
From the assumptions, we know 0 01 10 The Lyapunov number is given by

The impact of harvesting.
Here we discuss the influence of the harvesting function on the dynamics of the model. We compare our model (1)  With no harvesting, 0 P is always an unstable node, and 1 P is a saddle or a stable node. Whereas with harvesting, 0 P can also be a saddle, 1 P can also be unstable node.
Coexistence equilibria. The coexistence equilibria were found to be nodes, saddles or focus. By including harvesting, the coexistence equilibria in our model can be nodes, focus, or even of center type.
Bifurcations. The system with no harvesting undergoes some bifurcations: transcritical, saddle-node bifurcations. Our model has even richer dynamics, not only those bifurcations, but more importantly, Hopf bifurcations.