Global dynamics of three species omnivory models with Lotka-Volterra interaction

In this work, we consider the community of three species food web model with Lotka-Volterra type predator-prey interaction. In the absence of other species, each species follows the traditional logistical growth model and the top predator is an omnivore which is defined as feeding on the other two species. It can be seen as a model with one basal resource and two generalist predators, and pairwise interactions of all species are predator-prey type. It is well known that the omnivory module blends the attributes of several well-studied community modules, such as food chains (food chain models), exploitative competition (two predators-one prey models), and apparent competition (one predator-two preys models). With a mild biological restriction, we completely classify all parameters. All local dynamics and most parts of global dynamics are established corresponding to the classification. Moreover, the whole system is uniformly persistent when the unique coexistence appears. Finally, we conclude by discussing the strategy of inferior species to survive and the mechanism of uniform persistence for the three species ecosystem.


1.
Introduction. In this work, we consider the following three species food web model where all parameters are nonnegative real constants. In the absence of other species, each species follows the traditional logistic population growth with birth rates, r U ,  Figure 1. The diagram of Three species intraguild predator models is illustrated and each species has its own nutrient resource.
r V , r W , and environmental carrying capacities, K U , K V , K W , for the species U , V , W , respectively. And the nonlinear interactions between species are Lotka-Volterra type with omnivory which means the top predator (intraguild-predator) W are feeding on two resources, intraguild-prey V and prey U [6]. Biologically, we assume that all coefficients of interactions a ij are non-negative, and a ij is the rate of consumption for i < j or measures the contribution of the victim (resource or prey) to the growth of the consumer for i > j [10]. System (1) can be regarded as a food-chain model, a two predators-one prey model or a two preys-one predator model when a 13 = a 31 = 0, a 23 = a 32 = 0 or a 12 = a 21 = 0, respectively. Please refer Figure 1. It is well known that system (1) blends the attributes of several well-studied community modules, such as food chains, exploitative competition (two predators-one prey) and apparent competition (one predator-two preys) [6]. The most important feature of system (1) is involved omnivory which is believed that this property is crucial to to the stability of food web structure and its global dynamics.
Re-examine previous known three species food web model with omnivory [6,12,13,8,7], for the intermediate predator (intraguild prey) there is only one nutrient resource from the basal prey. However, in system (1) each species has its own nutrient resource governed by the logistic growth terms. Moreover, they affect each others weakly by the nonlinear terms. So system (1) can be seen as a type of three species food web system with diversity of food resources and weakly effects to each others. We think that these features appear in common situations. Our main purpose of this work is to answer what is the best strategy for each species to survive and what is the condition of uniform persistence for the whole system. The rest of the paper is organized as follows. In Section 2, we first show the boundedness of solutions of (1). Then local stability of all boundary equilibria are investigated by the linear method. Moreover, global dynamics of some boundary equilibria are showed by the methods of Lyapunov and Butler-McGehee Lemma. Next, with assumption (A) we classify all parameters to investigate the existence of positive equilibrium and its local dynamics numerically. In the final section, numerical evidences are presented, and some biological discussions and implications are given. linear method. Then some global dynamics are investigated by differential inequalities coupling with LaSalle's invariant principle and McGehee Lemma. Finally, we summarize a table which could completely classify all dynamics by the parameters.
To simplify the arguments, we apply the following scaling transformation to (1), then we obtain a simplified ODE model, Lemma 2.1. Solutions of (3) with nonnegative (positive) initial conditions are nonnegative (positive). Moreover, all solutions of (3) are bounded.
Proof. By Theorem 3 in [9], we know that solutions of (3) is bounded. It is also easy to see that x-axis, y-axis, z-axis, xy-plane, xz-plane, and yz-plane are invariant subspaces of (3). Hence, one can easily show that solutions with nonnegative (positive) initial conditions are nonnegative (positive) by the uniqueness of solutions.
Throughout this work, we always assume that (A): r U > a 12 K V or b > 1. Assumption (A) is actually a biological restriction which means that species U can sustain the negative effect with maximal amount of species V . Since it is easy to see that if assumption (A) does not hold then species U will die out eventually in the two-dimensional subsystem without species W . So this hypothesis can keep interest and complexity of system (3).
2.1. Existence of boundary equilibria and its local stability. In this subsection, we will find all corresponding conditions to establish the existence of boundary equilibria and their local stabilities.
By direct computations, we have the Jacobian matrix of system (3) is given by All boundary equilibria can be easily found and their Jacobian matrix are considered as follows.
The matrix J(E x ) has two positive eigenvalues and one negative eigenvalue. Clearly, E x is a saddle with one-dimensional stable manifold, the interior of the x-axis and two-dimensional unstable manifold with one tangent vector on the x-y plane and another one on the x-z plane.
(c) E y = (0, 1 b , 0). Direct computations imply that Since b > 1, the matrix J(E y ) has two positive eigenvalues and one negative eigenvalue. It follows that E y is a saddle with one-dimensional stable manifold, the interior of the y-axis and two-dimensional unstable manifold with one tangent vector on the x-y plane and another one on the y-z plane.
By the ordering of 1, c, and f , we state local stability of E z and omit the proof.
(i) If f < 1 and f < c, then J(E z ) has three negative eigenvalues and it follows that E z is stable; (ii) if f > 1 and f < c, then it is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the x coordinate and twodimensional stable manifold, the interior of the y-z plane; (iii) if f < 1 and f > c, then it is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the y coordinate and twodimensional stable manifold, the interior of the x-z plane; (iv) if f > 1 and f > c, E z is a saddle with one-dimensional stable manifold the interior of the z-axis and two-dimensional unstable manifold with one tangent vector on the x-z plane and another one on the y-z plane. (e) E xy = ( b−1 a+b , a+1 a+b , 0). Let ρ = b−1 a+b , then 1 − ρ = a+1 a+b . One can easily verify that J(E xy ) is of the following form Since b > 1, the equilibrium E xy exists. It is clear that J(E xy ) has at least one positive eigenvalue r z [1 + dρ + e(1 − ρ)]. Consider another two eigenvalues, λ 2 and λ 3 , of (4). They are actually the eigenvalues of the up-left 2 × 2 submatrix of (4). By direct computations, we obtain Hence it is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the z coordinate and two-dimensional stable manifold, the interior of the x-y plane.
The equilibrium E xz can exist only if f > 1. Clearly, J(E xz ) has one eigenvalue ). Consider another two eigenvalues, λ 2 and λ 3 , of (5). They are actually the eigenvalues of the 2 × 2 submatrix of (5) by removing the second column and the second row. By direct computations, we have then J(E xz ) has three negative eigenvalues and it follows that E xz is stable; and if then J(E xz ) has one positive eigenvalue and two negative eigenvalues. Similarly, E xz is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the y coordinate and two-dimensional stable manifold, the interior of the x-z plane.
Similarly the equilibrium E yz can exist only if f > c. Clearly, J(E yz ) has one eigenvalue Consider another two eigenvalues, λ 2 and λ 3 , of (8). They are actually the eigenvalues of the low-right 2 × 2 submatrix of (8). By direct computations, we obtain Then we have λ 2 < 0 and λ 3 < 0. Similarly, we find the following condition of stability for E yz . If

TING-HUI YANG, WEINIAN ZHANG AND KAIJEN CHENG
then E yz is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the x coordinate and two-dimensional stable manifold, the interior of the y-z plane. Here we summarize all local stability results for boundary equilibria in the following proposition.
Proposition 1. For system (3), the following statements are true.
The equilibrium E x always exists and is a saddle with one-dimensional stable manifold, the interior of the x-axis and two-dimensional unstable manifold with one tangent vector on the x-y plane and another one on the x-z plane. (iii) The equilibrium E y always exists and is a saddle with one-dimensional stable manifold, the interior of the y-axis and two-dimensional unstable manifold with one tangent vector on the x-y plane and another one on the y-z plane. (iv) The semi-trivial equilibrium E z always exists. And (a) if f < 1 and f < c then it is stable; (b) if f > 1 and f < c, then it is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the x coordinate and twodimensional stable manifold, the interior of the y-z plane; (c) if f < 1 and f > c, then it is saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the y coordinate and twodimensional stable manifold, the interior of the x-z plane; (d) if f > 1 and f > c, it is a saddle with one-dimensional stable manifold the interior of the z-axis and two-dimensional unstable manifold with one tangent vector on the x-z plane and another one on the y-z plane. (v) The boundary equilibrium E xy always exists. Moreover, it is a saddle point with one-dimensional unstable manifold with tangent vectors which are nonzero in the z coordinate and two-dimensional stable manifold, the interior of the x-y plane. (vi) The boundary equilibrium E xz exists if f > 1 and it is stable if (6) holds. If (7) holds, then E xz is a saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the y coordinate and two-dimensional stable manifold, the interior of the x-z plane. (vii) The boundary equilibrium E yz exists if f > c and it is stable if (10) holds.
Otherwise, if (11) holds, then E yz is a saddle with one-dimensional unstable manifold with tangent vectors which are non-zero in the x coordinate and twodimensional stable manifold, the interior of the y-z plane.

2.2.
Global dynamics of boundary equilibria. In this subsection, we investigate some global dynamics of boundary equilibria. By the foregoing subsection, we have the following conclusions: E 0 , E x , E y and E xy are unstable. So we will consider the other boundary equilibria, E z , E xz and E yz . For reader's convenience, in Table 1 we present all local and global dynamics which will be investigated in Section 2 and Section 3. First, we classify all parameters into two main categories, c < 1 and c > 1. Biologically, the parameter c = a 23 r U /(a 13 r V ) < 1 can be rewritten as the form which means that the species x is inferior to the species y in apparent competition [4]. By Proposition 1, we may further classify all parameters by the ordering of f , c and 1. Hence generically we consider the following six sub-cases: By the result (iv) of Proposition 1, E z is stable if f < min{1, c}. Actually, we can further show that it is globally asymptotically stable. This also clarify the global dynamics of cases (B)-1 and (C)-1. It is clear that E z is unstable for all other cases. Please refer "E z " column of Table 1. and E z is globally asymptotically stable.
Proof. By (3c), we have Let z(t) be the solution of the differential equation with the same initial condition of z(t). Then we have the following facts: So for any ε > 0, we can find a T > 0 such that for all t > T . Then x(t) converges to 0 as t tends to infinity. Finally, we consider the differential equation (3b): dy dt = r y y(1 − by) + r y axy − r y cyz.
for all t > T . So y(t) converges to 0 as t tends to infinity. Hence we can conclude that E z is globally asymptotically stable in the positive sector.
Biologically, the conditions f = r U a 13 K W < 1 and f = r U a 13 K W < c = a 23 r U a 13 r V can be rewritten as the form r U < a 13 K W and r V < a 23 K W , which imply that species x and y cannot sustain the negative effect with maximal amount of species z, then species x and y will become extinct eventually.
Next, we investigate global dynamics of equilibrium E xz . The equilibrium E xz can exist only if f = r U /(a 13 K W ) > 1. This can be seen that the species x can stand the exploitation of maximal amount of the species z. So this clarifies the cases of (B)-1, (B)-2, and (C)-1. Furthermore, by the foregoing discussion, the Jacobian matrix J(E xz ) has two negative eigenvalues and one eigenvalue, The following lemma says that equilibrium E xz is always saddle in the case of (B)-3.

The quantity af
However, it follows that E xz is stable if (6) holds. Consequently, we have the following global result which clarify partial global dynamics of cases (C)-2 and (C)-3. The complete dynamics of E xz can be found in the column E xz of Table 1.
Proof. Assumption (13) is equivalent to (1 + a)/d < (c − 1)/f . Hence we can take a positive number k such that (1 + a)/d < k < (c − 1)/f . Then consideṙ Therefore we have lim t→∞ y(t) = 0. Asymptotically, system (3) will approach the following two-dimensional subsystem, If we can show equilibrium E xz is GAS in the x-z plane, then we conclude that E xz is GAS in the positive octant of R 3 . Let E xz = (x,z) be the positive equilibrium, that is , 1 =x +z, and 1 = −dx + fz.

Consider the Lyapunov function
and by computation we obtain Then by LaSalle Invariant Principle, we can get that E xz is GAS in x-z plane. This completes the proof. (13) is a sufficient condition of (6).

Remark 1. It is clear that
For equilibrium E yz = (0, f −c bf +ce , b+e bf +ce ), it can exist only if f > c. So in cases of (B)-1, (C)-1 and (C)-2, E yz does not exist. It is easy to see that the inequality f > c is equivalent to Similarly, this inequality suggests that the species y can sustain the exploitation of maximal amount of the species z. If the equilibrium E yz exists, then its Jacobian matrix J(E yz ) has two negative eigenvalues and one eigenvalue, The following lemma says that equilibrium E yz is always saddle in the case of (C)-3. Lemma 2.3. In the case of (C)-3, the inequality bf + ce − b + c − e − f > 0 is always true, that is, (11) holds.
In the case of (B)-2, we always have and this implies E yz is stable. Moreover, we can prove the following global behavior. and equilibrium E yz is globally asymptotically stable.

Proof.
Considerẋ So we have lim t→∞ x(t) = 0. The following arguments are similar, so we omit them. Table 1. Existence and dynamics of equilibria by the classifications. The notations "Un" means unstable, " " means non-existence of equilibrium, "∃" means existence of equilibrium, and "GAS" means globally asymptotically stable.
With assumption af − cd + d + f < 0 In the case of (B)-3, if inequality (10) hold, then the equilibrium E yz is stable. Moreover, we have the following global result which clarifies partial dynamics of (B)-3.
Proposition 5. For case of (B)-3, assume that then E yz is globally asymptotically stable.
Proof. Assumption (15) is equivalent to c/b + f /e < 1/b + 1/e. Hence we can take a positive number k such that c/b + f /e < k < 1/b + 1/e. Then consider Therefore we have lim t→∞ x(t) = 0. The remaining arguments are similar, so we omit them.

Remark 2.
It is clear that (15) is a sufficient condition of (10).
Finally, we summarise all results in Table 1. 3. Existence of positive equilibrium and uniform persistence. In this section, we first find the necessary and sufficient conditions to guarantee the existence of positive equilibrium E * = (x * , y * , z * ). Then the condition of local stability of E * is presented by the Routh-Hurwitz Criterion. Although we cannot show the globally asymptotically stability of E * analytically, we can verify the system (3) is uniformly persistent when E * exists. In cases of (B)-1, (B)-2 and (C)-1, the global dynamics of (3) is classified in Section 2. So it is easy to see that E * does not exist in this three cases (Please refer Table 1). Therefore, we investigate the other cases in this section. To find positive equilibrium E * = (x * , y * , z * ) is equivalent to find the solution (x * , y * , z * ) of the linear system , with 0 < x * , y * , z * < 1. Here are the necessary and sufficient conditions for the existence of the positive equilibrium E * .
Proposition 6. Let assumption (A) hold. The coexistence equilibrium E * exists if and only if (7) and (11) hold.
Proof. Assume that the positive equilibrium E * = (x * , y * , z * ) exists, that is, there are three positive real numbers, x * , y * and z * , less than 1 and satisfying (16). By straightforward computation of system (16), we get the explicit formulations of solution (x * , y * , z * ), Since z * > 0 and ae + bd + a + b − d + e = ae + d(b − 1) + a + b + e > 0, we have ae+af +bd+bf −cd+ce > 0 by (17c). Therefore we also have bf +ce−b+c−e−f > 0 and af − cd − a − c + d + f > 0, that is, (7) and (11) hold. We complete the proof of this implication. For the other implication, we assume that (7) and (11) hold, that is, bf + ce − b + c − e − f > 0 and af − cd − a − c + d + f > 0. Then by adding these two inequalities, we obtain Consider the determinant of the linear system (16), by assumption (A). So the solution of system (16) exists, and has the form Finally, it can clearly be seen that 0 < x * , y * , z * < 1. This show the existence of E * . We complete the proof.
Hence E * exists. We summarize the existence results of E * in the column "E * " of Table 1.

TING-HUI YANG, WEINIAN ZHANG AND KAIJEN CHENG
(iv) The local stability of E * can be verified by Routh-Hurwitz criterion. The computations are tedious, so we put it in the Appendix. By observing the form, it suggests that E * is stable whenever it exists. But we cannot prove that. Some numerical simulations are discussed in the last section.
Please refer Table 1. The method is similar, so we only investigate case (i). It is easy to check that system (3) is persistent by the results of [3]. Our strategy is to use the main results in [1,2] to verify the uniform persistence of (3). It is sufficient to show that the boundary of the first octant for the solution of (3) is isolated and acyclic.
Under assumptions (A), (B)-3 and (11), the isolated invariant sets of solutions on the boundary are {E 0 , E x , E y , E z , E xy , E xz , E yz }. All possible chain from E 0 to other semi-trivial equilibria can been found for six cases : We only consider the first case, and the other cases are similar. If E 0 → E x → E xy happens, then it is clear that E xy can not be chained to E 0 or E x by Proposition 1 (v). Thus, the set of equilibria, on the boundary is acyclic and the system (3) is uniformly persistent.

4.
Discussions. In this work, we consider the community of three species food web model with Lotka-Volterra type predator-prey interaction. Each species has its own nutrient resource governed by the traditional logistical growth. And they affect each other by the interplay of competition and predation. In particular, the top predator is an omnivore which is defined as feeding on the other two species. With a mild biological restriction (A) we have classified all parameters and investigated their corresponding dynamics which are summerized in Table 1.
First, in case (B)-1 and case (C)-1, we showed that species U and V die out and W survives. Since the inequalities f < 1 and f < c represent that species U and V cannot stand the exploitation by species W in the following equivalent forms, r U < a 13 K W and r V < a 23 K W , respectively. Hence E z is globally asymptotically stable.
In Section 2, we have classified all parameters into two main categories, c < 1 and c > 1. Biologically, the parameter c = a 23 r U /(a 13 r V ) can be rewritten as the form where the ratio r U /a 13 means the birth-rate of U overs consuming rate a 13 by predator W and the ratio r V /a 23 means the birth-rate of V overs consuming rate a 23 by predator W . Hence assumption c < 1(c > 1) can be interpreted that species U is inferior (superior) to species V under the apparent competition [4]. So in the category (B), any equilibrium involved species U is unstable or does not exist except for the case of (B)-3 with (11). Similarly, in the category (C), any equilibrium involved species V is unstable or does not exist except for the cases (C)-2 and (C)-3) with (7). This three exceptions are exactly cases where E * exists and the system uniformly persists. We will discuss in more detail later. Next, in case (B)-2, we showed that species U dies out, and V , W survives, since V can sustain the exploitation by W , because of f > c (r V > a 23 K W ). In addition, species U lost the apparent competition. Hence we have the globally asymptotical stability of E yz .
Let us discuss the most interesting and complex cases, (B)-3, (C)-2 and (C)-3. In the case of (B)-3, that is c < 1 < f , inequalities f > c and f > 1 imply that species U and species V can sustain the exploitation of maximal amount of species W , respectively. But c < 1 means that species U is inferior to species V in apparent competition. How does species U survive? The inequality (7) can be rewritten as the form, In right hand side, the only positive term is b(f − 1). So the only possibility to make (7) true if b = r U /(a 12 K V ) is large enough. Either species U take r-strategy or the amount of species V is small. So in the case of (B)-3 with (7), species U can survive and E * exists.
For the case of (C)-2, that is 1 < f < c, inequalities f > 1 and f < c represent that species U can sustain the exploitation of maximal amount of species z, but species y cannot. Moreover, the inequality c > 1 means species V lost the apparent competition. Similarly, how does species V survive? The inequality (11) can be rewritten as the form, In right hand side, the only positive term is a(f − 1). The only possibility to make (11) true if a = a 21 K U /r V is large enough. The possible strategy for species V to survive is to improve the efficiency of consuming species U . Hence in the case of (C)-2 with (11), species V can survive and E * exists. For the case of (C)-3, that is 1 < c < f , species U and species V can stand the exploitation of maximal amount of species W , but species V lost the apparent competition. Similarly, in the right hand side of (19), there are two positive terms, a(f − 1) and (f − c). There are possible strategies for species V . One is to improve the efficiency of consuming species U , and another one is r-strategy.
Finally, we try to answer the questions which we propose, what is the best strategy for each species to survive and what is the condition of uniform persistence for the whole system. For species U , to survive in any cases discussed above is r-strategy. And for species V the best strategy is to improve the efficiency of consuming rate.
Appendix. In this appendix, we investigate the local stability of the coexistence equilibrium E * . The Jacobian matrix evaluated at E * = (x * , y * , z * ) is J(x * , y * , z * ) =   −r x x * −r x x * −r x x * ar y y * −br y y * −cr y y * dr z z * er z z * −f r z z *   .
By direct computations, the characteristic polynomial of J(x * , y * , z * ) is