Dynamics and Patterns of a Diffusive Prey-Predator System with a Group Defense for Prey.

1. Introduction

The predator-prey system first proposed by [1,2] is one of the fundamental ecological systems in both ecology and mathematical ecology. Based on different settings, various types of predator-prey models described by differential systems have been proposed and the dynamics of these systems are studied [3-6]. The basic form of these models is as follows:

[mathematical expression not reproducible], (1)

where r is the intrinsic growth rate and K is the environmental carrying capacity of prey population, and the function P(x) is the functional response; the constant c(>0) is the ratio of biomass conversion and s is the natural death rate of predator species. The simplest functional response is Lotka-Volterra function which is described as

[mathematical expression not reproducible], (2)

which is also called Holling type I function. However, the curve defined by the Lotka-Volterra response function is a straight line through the origin and is unbounded. Thus, more reasonable response functions should be nonlinear and bounded. In 1913, Michaelis and Menten proposed the response function

P(x, y) = [mxy/a + x], (3)

where m >0 denotes the maximal growth rate of the species and a > 0 is the half-saturation constant. It is now referred to as a Michaelis-Menten function or a Holling type II function. Another class of response function is

P(x, y) = [m[x.sup.2]y/a + bx + [x.sup.2]] (4)

which is called a sigmoidal response function, while the simplification

P(x, y) = [m[x.sup.2]y/a + [x.sup.2]] (5)

is known as a Holling type III function. Some authors [7, 8] considered system (1) with following response function:

P(x, y) = [mxy/a + [x.sup.2]] (6)

which is called Holling type IV function. Besides, Beddington-DeAngelis type P(x,y) = [beta]xy/([alpha] + x + my) and more complicated functional response P(x, y) = [beta][x.sup.2]y/([x.sup.2] + m[y.sup.2]) are also considered by some researchers [9, 10].

Recently, some works consider the case when animals join together in herds in order to provide a self-defense from predators. In , the authors argued that it is more appropriate to model the response functions of prey that exhibit herd behavior in terms of the square root of the prey population. Inspired by this thought, the authors in  choose response function P(x) = [square root of x] to reflect this fact. When motion is allowed,  considered the spatiotemporal behavior of a prey-predator system with a group defense for prey by means of extensive computer simulations. The proposed model is as follows:

[mathematical expression not reproducible], (7)

where u and v denote, respectively, the densities of prey and predator species. r is the growth rate of prey species, K is its carrying capacity, [r.sub.2] is the mortality rate of predator species, [rho] is the search efficiency of predator for prey, [beta] is the biomass conversion coefficient, and [alpha] [member of] (0, 1) represents a kind of aggregation efficiency. The local dynamics for nonspatial model was studied, such as Hopf bifurcation and existence of extinction domain. For model (7), the authors only give some numerical simulations to find some spatiotemporal features. Reference  considers the direction and the stability of the bifurcating periodic solutions for model (7) with [alpha] = 1/2 under Neumann boundary conditions. Reference  investigated the global dynamics of nonspatial model including the nonexistence of periodic orbits and the existence and uniqueness of limit cycles. We refer readers to [16-21] as some other related works on predator-prey model with herd behavior.

It is noted that up to now no one has studied the existence and nonexistence of positive steady state solutions of (7). Therefore, the main aim of this article is to study the existence and nonexistence of nonconstant positive solutions of the following elliptic system:

[mathematical expression not reproducible], (8)

where v is the outward unit normal vector on [partial derivative][OMEGA], and we impose a homogeneous Neumann type boundary condition, which implies that (8) is a closed system and has no flux across the boundary [partial derivative][OMEGA].

The structure of this paper is arranged as follows. In Section 2, we estimate the a priori bounds of positive solutions of (7). In Section 3, the local and global stabilities of nonnegative constant steady states of (7) are discussed. In Section 4, we give a priori estimate for the positive solutions of (8) by using maximum principle and Harnack inequality. In Section 5, we give a nonexistence result of nonconstant solutions of (8). In Section 6, we consider the existence of nonconstant positive solutions of (8). Finally, to support our theoretical predictions, some numerical simulations are given.

2. Basic Dynamics and a Priori Bound

Theorem 1. For system (7), one has the following.

(a) If [u.sub.0](x) [greater than or equal to] 0, [v.sub.0](x) [greater than or equal to] 0, then system (7) has a unique solution (u(t, x), v(t, x)) such that u(t, x) > 0, v(t, x) > 0 for t [member of] (0, +[infinity]) and x [member of] [bar.[OMEGA]].

(b) Any solution (u, v) of (7) satisfies

[mathematical expression not reproducible]. (9)

Proof. (a) Define

[mathematical expression not reproducible]. (10)

Then [f.sub.v] = -[rho][u.sup.[alpha]] [less than or equal to] 0 and [g.sub.u] = [beta][alpha][u.sup.[alpha]-1]v [greater than or equal to] 0 in [bar.[R.sup.2.sub.+] = [u [greater than or equal to] 0, v [greater than or equal to] 0}. Hence, (7) is a mixed qusi-monotone system. Consider following system:

[mathematical expression not reproducible]. (11)

Assume u(t; [u.sub.0], [v.sub.0]), v(t; [u.sub.0], [v.sub.0]) are the unique solution to system (11). Let

[mathematical expression not reproducible]. (12)

Obviously, ([u.bar](t, x), [v.bar](t, x)) = (0, 0) and ([bar.u](t), [bar.v](t)) = (u(t; [[phi].sub.M], [[psi].sub.M]), v(t; [[phi].sub.M], [[psi].sub.M])) are a pair of lower-solution and upper-solution to system (7). Therefore, according to the Theorem 8.3.3 in  or Theorem 5.3.2 in , system (7) has a unique globally defined solution (u(x, t), v(x, t)) which satisfies

[mathematical expression not reproducible]. (13)

The strong maximum principle implies that u(x, t), v(x, t) > 0 when t >0 for all x [member of] [bar.[OMEGA]].

(b) By the first equation of (7), we easily obtain the fact that [r.sub.1]u(1-u/K)-[rho][u.sup.[alpha]]v [less than or equal to] [r.sub.1]u(1-u/K) in [0, +[infinity])x[OMEGA]; the first result follows easily from the simple comparison argument for parabolic problems, and thus there exists T [member of] (0, +[infinity]) such that u(t, x) [less than or equal to] K + [epsilon] in [T, +[infinity]) x [OMEGA] for an arbitrary constant [epsilon] > 0.

For the estimate of [mathematical expression not reproducible]; then

[mathematical expression not reproducible], (14)

[mathematical expression not reproducible]. (15)

Multiplying (14) by [beta]/[rho] and adding it to (15), we have

[mathematical expression not reproducible]. (16)

Integration of the inequality leads to

[mathematical expression not reproducible]. (17)

3. Stability of the Nonnegative Constant Steady States of (7)

In this section, we will analyze the stability of nonnegative constant steady states of (7). By the direct computation, we see that the possible nonnegative constant steady states of (7) are

[mathematical expression not reproducible], (18)

where [u.sup.*] = [([r.sub.2]/[beta]).sup.1/[alpha]], [v.sup.*] = [r.sub.1][u.sup.*(1-[alpha])] (1 - [u.sup.*]/K)/[rho]. Obviously, the positive constant steady state [E.sup.*] exists if [r.sub.2] < [beta][K.sup.[alpha]] holds.

Notation 1. Let 0 = [[mu].sub.0] < [[mu].sub.1] < [[mu].sub.2] < ... < ... [right arrow] [infinity] be the eigenvalues of -[DELTA] on [OMEGA] under homogeneous Neumann boundary condition. We define the following space decomposition:

(i) S([[mu].sub.n]) is the space of eigenfunctions corresponding to [[mu].sub.i] for n = 0, 1, 2,....

(ii) [X.sub.ij] := {c x [[phi].sub.ij] : c [member of] [R.sup.2]}, where {[[phi].sub.ij]} are orthonormal basis of S([[mu].sup.n]) for j = 1, 2, ..., dim[S([[mu].sub.n])].

(iii) [mathematical expression not reproducible], and so [mathematical expression not reproducible].

Let [??] be a nonnegative constant steady state of (7); then the linearization of (7) at a constant solution [??] can be expressed by

[u.sub.t] = (D[DELTA] + J) u, (19)

where D = diag([d.sub.1], [d.sub.2]), u = [(u(x, t), v(x, t)).sup.T], and

[mathematical expression not reproducible]. (20)

In view of Notation 1, we can induce the eigenvalues of system (19) confined on the subspace [X.sub.i]. If [lambda] is an eigenvalue of (19) on [X.sub.i], it must be an eigenvalue of the matrix -[[mu].sub.n]D + J for each n [greater than or equal to] 0. It is easy to see that A satisfies the characteristic equation

[mathematical expression not reproducible]. (21)

Theorem 2. (i) The trivial equilibrium [E.sub.0] = (0,0) is unstable.

(ii) If [beta][K.sup.[alpha]] < [r.sub.2], then [E.sub.1] = (K, 0) is globally asymptotically stable.

(iii) If 1 - [alpha] - ((2 - [alpha])/K)[u.sup.*] < 0, then [E.sup.*] is locally asymptotically stable.

Proof. (i) For [E.sub.0] = (0, 0), the eigenvalues are

[mathematical expression not reproducible]. (22)

Obviously, [E.sub.0] is unstable.

(ii) For [E.sub.1] = (K, 0), the eigenvalues are

[mathematical expression not reproducible]. (23)

If [beta][K.sup.[alpha]] < [r.sub.2], then [[lambda].sub.1n] and [[lambda].sub.2n] are all negative. Therefore [E.sub.1] is locally asymptotically stable. Indeed, [E.sub.1] is globally asymptotically stable.

On account of Theorem 1, we have lim [sup.sub.t[right arrow]+[infinity]] [max.sub.[bar.[OMEGA]]] u(*, t) [less than or equal to] K, and thus there exists [T.sub.1] [member of] (0, +[infinity]) such that, for an arbitrary constant 0 < [epsilon] < [([r.sub.2]/[beta]).sup.1/[alpha]] - K,

u(*, t) [less than or equal to] K + [epsilon], t [greater than or equal to] [T.sub.1]. (24)

It follows from the second equation of (7) that

[v.sub.t] - [d.sub.2][DELTA]v [less than or equal to] v [([beta](K + [epsilon]).sup.[alpha]] - [r.sub.2]), t [greater than or equal to] [T.sub.1]. (25)

Therefore, lim [sup.sub.t[right arrow]+[infinity]] [max.sub.[bar.[OMEGA]]] v(*, t) [less than or equal to] 0, and there exists [T.sub.2] > [T.sub.1] such that

v (*, t) [less than or equal to] [epsilon], t [greater than or equal to] [T.sub.2]. (26)

It follows from the first equation of (7) that

[mathematical expression not reproducible]. (27)

On account of and the arbitrariness of [epsilon] > 0, we have lim [inf.sub.t[right arrow]+[infinity]] [min.sub.[bar.[OMEGA]]]u(*, t) [greater than or equal to] K. This combined with lim [sup.sub.t[right arrow]+[infinity]] [max.sub.[bar.[OMEGA]]]u(*, t) [less than or equal to] K allows us to derive

[mathematical expression not reproducible]. (28)

Hence, [E.sub.1] is globally asymptotically stable when [r.sub.2] > [K.sup.[alpha]][beta].

(iii) When [E.sup.*] = ([u.sup.*], [v.sup.*]) exists, the corresponding characteristic equation is as follows:

[mathematical expression not reproducible]. (29)

Obviously, we have

[mathematical expression not reproducible]. (30)

If 1 - [alpha] - ((2 - [alpha])/K)[u.sup.*] < 0, then [[lambda].sub.1n] + [[lambda].sub.2n] < 0 and [[lambda].sub.1n][[lambda].sub.2n] > 0. Hence, all the roots of (29) have negative real part which means that [E.sup.*] is locally asymptotically stable when 1 - [alpha] - ((2 - [alpha])/K)[u.sup.*] < 0.

4. The Prior Estimate

In this section, we will give some a priori estimates of positive solutions to (8). Firstly, we give two known lemmas.

Lemma 3 (Harnack inequality (cf. )). Let [omega] [epsilon] [C.sup.2]([OMEGA]) n [C.sup.1]([bar.[OMEGA]]) be a positive classical solution to

[mathematical expression not reproducible]. (31)

Then there exists a positive constant C such that

[mathematical expression not reproducible]. (32)

Lemma 4 (maximum principle (cf. )). Suppose that g [member of] C([bar.[OMEGA]] x R).

(i) Assume that [omega] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]) satisfies

[mathematical expression not reproducible]. (33)

If [omega]([x.sub.0]) = [max.sub.[bar.[OMEGA]]][omega], then g([x.sub.0], [omega]([x.sub.0])) [greater than or equal to] 0.

(ii) Assume that [omega] [member of] [C.sup.2]([OMEGA]) [intersection] [C.sup.1]([bar.[OMEGA]]) satisfies

[mathematical expression not reproducible]. (34)

If [omega]([x.sub.0]) = [min.sub.[bar.[OMEGA]]][omega], then g([x.sub.0], w([x.sub.0])) [less than or equal to] 0.

Lemma 5. For any positive solution (u, v) of system (8),

[mathematical expression not reproducible]. (35)

for any x [member of] [bar.[OMEGA]].

Proof. Form Lemma 4, u(x) [less than or equal to] K and from the strong maximum principle u(x) < K for all x [epsilon] [bar.[OMEGA]]. Multiplying the first equation of (8) by [beta]/[rho] and adding it to the second equation, we have

[mathematical expression not reproducible]. (36)

Then the maximum principle implies that

[mathematical expression not reproducible]. (37)

Hence, v(x) < ([beta]/[rho])([d.sub.1]/[d.sub.2] + [r.sub.1]/[r.sub.2])K.

In the following, we estimate the positive lower bound of positive solution of (8).

Theorem 6. Let [OMEGA] be a bounded smooth domain in [R.sup.n]. There exist two positive constants [C.bar] < [bar.C] depending possibly on [d.sub.1], [d.sub.2], K, [beta], [alpha], [rho], and [OMEGA], such that such that any positive solution (u(x), v(x)) of system (8) satisfies

[mathematical expression not reproducible] (38)

for any x [epsilon] [bar.[OMEGA]].

Proof. From Lemma 5, we obtain

[mathematical expression not reproducible], (39)

where [bar.C] depends on [d.sub.1], [d.sub.2], K, [beta], [alpha], and [rho].

From Lemma 3, we obtain the fact that there exists a positive constant [C.sub.2] such that

[mathematical expression not reproducible]. (40)

On the contrary, suppose the result is false. Then there exists a sequence {([u.sub.n], [v.sub.n])} of positive solutions to system (8) such that

[mathematical expression not reproducible]. (41)

By the regularity theory for elliptic equations, there exists a subsequence of {([u.sub.n], [v.sub.n])}, which will be denoted again by {([u.sub.n], [v.sub.n])}, such that {([u.sub.n], [v.sub.n])} [right arrow] ([u.sub.0], [v.sub.0]) in [C.sup.2]([bar.[OMEGA]]) as n [right arrow] +[infinity]. Observe that [u.sub.0] [less than or equal to] K and, from (41), either [u.sub.0] [equivalent to] 0 or [v.sub.0] [equivalent to] 0. Therefore, we have the following two cases:

(i) [u.sub.0] [equivalent to] 0, [v.sub.0] [not equivalent to] 0; or [u.sub.0] [equivalent to] 0, [v.sub.0] [equivalent to] 0.

(ii) [u.sub.0] [not equivalent to] 0, [v.sub.0] [equivalent to] 0.

Since {([u.sub.n], [v.sub.n])} is a positive solution of (8), one can obtain the following integral equation by integrating (8) for [u.sub.n] and [v.sub.n] over [OMEGA]:

[mathematical expression not reproducible]. (42)

(i) In this case, [u.sub.0] [equivalent to] 0; then

[beta][u.sup.[alpha].sub.n] - [r.sub.2] [right arrow] -[r.sub.2] < 0 (43)

uniformly as n [right arrow] [infinity] and [v.sub.n] > 0; then for sufficiently large n, we have

[mathematical expression not reproducible], (44)

which is a contradiction.

(ii) If [u.sub.0] [not equivalent to] 0, [v.sub.0] [equivalent to] 0, then this implies that [u.sub.0] satisfies (8). So [u.sub.0] [equivalent to] K for large n. Thus

[beta][u.sup.[alpha].sub.n] - [r.sub.2] [right arrow] [beta][K.sup.[alpha]] - [r.sub.2] > 0 (45)

for large n since [beta][K.sup.[alpha]] < [r.sub.2], which derives a contradiction again to the second integral equation of (42). This completes the proof.

5. Nonexistence of Nonconstant Positive Steady States

In this section, we can show the nonexistence of nonconstant positive solutions to system (8) when the diffusion coefficients [d.sub.1] and [d.sub.2] are large.

Theorem 7. There exists a positive constant [d.sup.*] such that elliptic problem (8) has no nonconstant positive solution if min{[d.sub.1], [d.sub.2]} > [d.sup.*].

Proof. Suppose that (u(x), v(x)) is a nonconstant positive solution of system (8). Denote [mathematical expression not reproducible]. Then

[mathematical expression not reproducible]. (46)

Define

H (u) = [[bar.u] - u/[[bar.u].sup.[alpha]] - [u.sup.[alpha]] (47)

for [bar.u] [not equal to] u. Indeed, we can prove that h(u) > 0 and h'(u) > 0. In fact, notice

[mathematical expression not reproducible]. (48)

Let

[h.sub.1] (u) = (1 - [alpha]) [u.sup.[alpha]] + [alpha][bar.u][u.sup.([alpha]-1)] - [[bar.u].sup.[alpha]], (49)

and we have

[h'.sub.1] (u) = [alpha] (1 - [alpha]) [u.sup.([alpha]-2)] (u - [bar.u]), (50)

which implies that [h.sub.1] (u) > min [h.sub.1](u) = [h.sub.1] ([bar.u]) = 0 for u [not equal to] [bar.u]. Therefore, we obtain the fact that h(u) > 0.

Furthermore, multiplying the first equation of (8) by [beta]/[rho], adding it to the second equation, and integrating over [OMEGA], we get

[mathematical expression not reproducible], (51)

and then the Neumann boundary conditions lead to

[mathematical expression not reproducible]. (52)

Thus

[mathematical expression not reproducible]. (53)

Multiplying the first equation in (8) by u - [bar.u], we have

[mathematical expression not reproducible]. (54)

Multiplying the second equation in (8) by v - [bar.v], we have

[mathematical expression not reproducible]. (55)

From (54) and (55) and the Poincare inequality, we obtain

[mathematical expression not reproducible], (56)

where

[mathematical expression not reproducible]. (57)

Hence, if

min {[d.sub.1], [d.sub.2]} > [d.sup.*] := [1/[[mu].sub.1]] max {A, B}, (58)

then

[nabla] (u - [bar.u]) = [nabla] (v - [bar.v]) = 0, (59)

and (u, v) must be a constant solution.

6. Existence of Nonconstant Positive Steady States

In this subsection, we discuss the existence of nonconstant positive solutions to system (8) when the diffusion coefficients [d.sub.1] and [d.sub.2] vary while the parameters [r.sub.1], K, [alpha], [rho], [beta], and [r.sub.2] are fixed by using the Leray-Schauder degree theory. Throughout this section, we assume that the positive constant steady state [E.sup.*] = ([u.sup.*], [v.sup.*]) exists.

For simplicity, denote u = (ii, v) and

[mathematical expression not reproducible]. (60)

Thus, (8) can be written as

[mathematical expression not reproducible], (61)

and, obviously, u is a positive solution of (61) if and only if

F (u) := u - [(I - [DELTA]).sup.-1] ([D.sup.-1][PHI](u) + u) in [X.sup.+], (62)

where [(I - [DELTA]).sup.-1] is the inverse of I - [DELTA] with the homogeneous Neumann boundary condition. As F(*) is a compact perturbation of the identity operator, the Leray-Schauder degree deg(F(*), [LAMBDA], 0) is well-defined from Theorem 6. By direct computation, we have

[F.sub.u] ([E.sup.*]) = I - [(I - [DELTA]).sup.-1] ([D.sup.-1][[PHI].sub.u] ([E.sup.*]) + I). (63)

If [F.sub.u]([E.sup.*]) is invertible, the index of F is defined as

index (F (*), [E.sup.*]) = [(-1).sup.[gamma]], (64)

where [gamma] is the number of negative eigenvalues of [F.sub.u]([E.sup.*]). Note that [lambda] is an eigenvalue of [F.sub.u]([E.sup.*]) on [X.sub.j] if and only if it is an eigenvalue of the matrix

[mathematical expression not reproducible]. (65)

Thus [F.sub.u]([E.sup.*]) is invertible if and only if, for all j [greater than or equal to] 0, the matrix [B.sub.j] is nonsingular. Writing

[mathematical expression not reproducible], (66)

we have that if H([d.sub.1], [d.sub.2]; [mu]) [not equal to] 0, then H([d.sub.1], [d.sub.2]; [mu]) < 0 if and only if the number of negative eigenvalues of [F.sub.u]([E.sup.*]) in [X.sub.j] is odd. The following lemma gives the explicit formula of calculating the index.

Lemma 8. If H([d.sub.1], [d.sub.2]; [[mu].sub.i]) = 0 for all i [greater than or equal to] 0, then index (F (*), [E.sup.*]) = [(-1).sup.[gamma]],

[mathematical expression not reproducible], (67)

where m([[mu].sub.i]) is the algebraic multiplicity of [[mu].sub.i].

To facilitate our computation of deg(F(*), [E.sup.*]), we only need consider the sign of det[[mu]D - [[PHI].sub.u]([E.sup.*])]. The direct calculation gives

[mathematical expression not reproducible]. (68)

Obviously, nonnegative roots of (68) exist if and only if [a.sup.2.sub.1][d.sub.2] - 4[d.sub.1][a.sub.2][a.sub.3] > 0 and [a.sub.1] > 0. Assume that [[mu].sup.+] and [[mu].sup.-] are the two roots of (68), we have the following conclusion.

Theorem 9. Assuming that [beta][K.sup.[alpha]] > [r.sub.2],

[mathematical expression not reproducible], (69)

and there exist i, j [member of] N, such that 0 [less than or equal to] [[mu].sub.j] < [[mu].sup.-] < [[mu].sub.j+1] [less than or equal to] [[mu].sub.i] < [[mu].sup.+] < [[mu].sub.i+1] and [[summation].sup.i.sub.k=j+1] m([[mu].sub.k]) is odd, then (8) has at least one nonconstant positive solution.

Proof. For t [member of] [0, 1], we define

[mathematical expression not reproducible], (70)

where [d.sup.*] is defined in Theorem 7.

The positive solutions of the problem

[mathematical expression not reproducible] (71)

are contained in [LAMBDA]. Note that u is a positive solution of system (8) if and only if it is a positive solution of (71) with t = 1. [u.sup.*] is the unique positive constant solution of (71) for any t [member of] [0, 1]. According to the choice of [d.sup.*] in Theorem 7, we have [E.sup.*] which is the only fixed point of [A.sub.0].

deg (I - [A.sub.0], [LAMBDA], 0) = index (I - [A.sub.0], [LAMBDA], [E.sup.*]) = 1. (72)

Since F = I - H(*, 1) and if (8) has no other solutions except the constant one [E.sup.*], then we have

[mathematical expression not reproducible]. (73)

On the other hand, by the homotopy invariance of the topological degree,

deg (I - [A.sub.0], [LAMBDA], 0) = deg (I - [A.sub.1], [LAMBDA], 0), (74)

which is a contradiction. Therefore, there exists at least one nonconstant solution of (8).

7. Numerical Simulation

7.1. Global Stability of Equilibrium [E.sub.1]. Consider system (7) with following parameters: [d.sub.1] = 0.8, [d.sub.2] = 0.9, [r.sub.1] = 0.9, [beta] = 0.1, [r.sub.2] = 0.2, K = 2, [rho] = 0.1, and [alpha] = 2/3. According to the discussions in Section 3, the steady state [E.sub.1] is globally asymptotically stable; see Figure 1.

7.2. Stability of Positive Steady State [E.sup.*]. Consider system (7) with following parameters: [d.sub.1] = 0.8, [d.sub.2] = 0.9, [r.sub.1] = 0.9, [beta] = 0.3, [r.sub.2] = 0.2, K = 2, [rho] = 0.1, and [alpha] = 2/3. According to the discussions in Section 3, the positive steady state [E.sup.*] is locally asymptotically stable; see Figure 2.

8. Conclusions

In this paper, we have investigated the existence/nonexistence of nonconstant positive steady states for a diffusive predator-prey system with a group defense for prey under Neumann boundary conditions. The existence results provide a theoretical support for pattern formation caused by diffusion. We also study the stability of nonnegative equilibria and obtain the fact that [E.sub.1] is globally asymptotically stable when [beta][K.sup.[alpha]] < [r.sub.2]. In fact, the positive steady state does not exist at this time. If [beta][K.sup.[alpha]] > [r.sub.2] and 1 - [alpha] - ((2 - [alpha])/K)[u.sup.*] < 0, then the positive steady state [E.sup.*] is locally asymptotically stable. It is easily obtained that when 1 - [alpha] - ((2 - [alpha])/K)[u.sup.*] = 0, characteristic equation (29) has a pair of purely imaginary roots. Therefore, system (7) occurs with Hopf bifurcation, as shown in Figure 3.

https://doi.org/10.1155/2018/6519696

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work is sponsored by Natural Science Foundation of Jiangsu Province (CN) (BK20150420).

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Honglan Zhu (iD) (1) and Xuebing Zhang (iD) (2)

(1) Business School, Huaiyin Institute of Technology, Huaian, China

(2) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China

Correspondence should be addressed to Xuebing Zhang; zxb1030@163.com

Received 11 September 2017; Accepted 26 November 2017; Published 8 January 2018

Academic Editor: Chris Goodrich

Caption: Figure 1: The steady state [E.sub.1] is globally asymptotically stable.

Caption: Figure 2: The positive steady state [E.sup.*] is locally asymptotically stable.

Caption: Figure 3: Hopf bifurcation occurs with parameters [r.sub.1] = 0.9, [alpha] = 2/3, [beta] = 0.1, [r.sub.2] = 0.18, K = 10; [rho] = 0.2.
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