Mathematical Analysis of an Epidemic-Species Hybrid Dynamical System

Abstract

We consider an epidemic-species hybrid dynamical system. The disease is spread among the prey only and the infected prey can reproduce virus. The predator only eats the infected prey. Mathematical analyses are given for the system with regard to the existence of equilibria, local stability, Hopf bifurcation, and the orbital stability of the Hopf bifurcating limit cycle. We further analyse the system under impulsive releasing of virus and predator.

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Academic Editor:Hai-Feng Huo

1, Department of Science and Technology, Guangxi University of Science and Technology, Liuzhou 545006, China

Received 9 March 2014; Revised 30 June 2014; Accepted 20 July 2014; 28 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Epidemic models and species models have received much attention from scientists, respectively. There are many literatures about them. Now we consider an epidemic-species hybrid dynamical system which is motivated by the integrated pest management (IPM). As we know, IPM is becoming more and more popular among farmers, researchers, and policy makers. IPM seeks to minimize reliance on pesticides by emphasizing the contribution of other control methods, including biological control, host-plant resistance breeding, and cultural techniques. Potentially, the use of viruses, fungi, and bacteria is one of the most effective biological methods for controlling pests. One reason for this optimism is that the generation time lapse of microbes is much shorter than that of an insect pest [1]. The role of microbial pesticides in the integrated management of insect pests has been recently reviewed for agriculture [2-5], forestry [6, 7], and public health [8]. In most cases no single microbial control agent will provide sustainable control of an insect pest or complex of pests. As components of an integrated approach, entomopathogens can provide significant and selective insect control. In the not too distant future we can envision a broader appreciation for the attributes of entomopathogens and expect to see synergistic combinations of microbial control agents with other technologies (in combination with semichemical technology, soft chemical pesticides, other natural enemies, resistant plants, chemigation, remote sensing, etc.) that will enhance the effectiveness and sustainability of integrated control strategies.

There is a vast amount of literature on the applications of microbial disease to suppress pests [9, 10]. But there are only a few papers on mathematical models of the dynamics of microbial disease in pest control [11, 12]. On the other hand, there are many mathematical models of infectious diseases of human beings [13-16]. All the earlier pest control models are normally two-dimensional consisting of susceptible and infected pests, and infectious disease models are at most three-dimensional with a susceptible, infected, and removable class; the dynamics of the diseases-causing organism is not considered in the model. However, presently the insect pathogens are used in two ways [17]. In the first method, a small amount of pathogens is introduced in the pest population with the expectation that it will generate an epidemic and will be subsequently endemic. The success of the approach depends on the survival of the microbes which in turn depends on environmental factors, such as temperature, humidity, and crop conditions. In the second method, an insect pathogen is used like a nonresidual chemical insecticide. In this case, it is applied whenever a pest population is at an economically extreme level for pest damage and there is no expectation that the pathogen will survive for an appreciable period. In this paper, we examine the use of pathogens in a four-dimensional prey-predator model for the agricultural or forestry ecosystem with the assumption that viral disease spreads only among the insect pests. In nature, the infection by baculovirus begins when an insect eats virus particles on a plant, perhaps from a sprayed treatment. Virus infection causes the cell lysis in the host body and produces more virus particles (virus replication), until the cell and ultimately the insect die. Most baculoviruses cause the host insect to die in a way that maximizes the chance that other insects come in contact with the virus and become infected by eating the foliage that has been contaminated by virus-killed larvae. We further assume that the natural enemy (predator) in the system survives on the infected prey. This is due to the fact that the viral infection makes some behavioral changes and sublethal effects on host, which make them more vulnerable to predation by natural enemies than healthy hosts before death [18].

The dynamical behavior of the considered system is investigated from the point of view of stability and persistence. The model shows that infection can be sustained only above a threshold force of infection resulting from virus replication parameter k . Still, on increasing the value of k , the endemic equilibrium bifurcates towards a periodic solution. The stability of the limit cycles arising from Hopf bifurcation is analysed using Poore's condition [19]. Whereas, in consideration of the practice in pest control, the natural enemy and the pathogens are introduced discontinuously at some fixed moment, the impulsive releasing of the virus and natural enemy is added in the system. We prove that the pest extinction periodic solution exists and is globally asymptotically stable when the impulsive period τ is less than the critical value; otherwise, the system can be permanent. The arrangement of the paper is as follows. In Section 2 we describe the model formulation. Section 3 shows its equilibria. Section 4 discusses the local stability and Hopf bifurcation. In Section 5 we give the details about the orbital stability of the periodic orbit arising from Hopf bifurcation by Poore's condition. In Section 6 we consider the corresponding impulsive system by releasing virus and natural enemy at some fixed moment. Lastly in Section 7 we end the paper with a concluding discussion.

2. The Model

The model contains three species, namely, pest, predator, and virus. The total pest population N is divided into two subpopulation classes: the susceptible pest, denoted by S , and the virus-infected pest, denoted by I . Therefore, at any time t , N(t)=S(t)+I(t) . We assume that only susceptible pest S is capable of reproducing with logistic law; that is, the infected class of pest is removed by lysis before having the possibility of reproducing. However, they still contribute with S to population growth towards the carrying capacity K . r is the intrinsic birth rate. Then the growth equation of susceptible pest S is given by [figure omitted; refer to PDF] A susceptible pest becomes infected as the two mentioned methods ahead. The incidence is assumed to be the simple mass action incidence λSV ; λ represents the effective per pest contact rate with viruses. Hence, the evolution equation of the susceptible pest is [figure omitted; refer to PDF] The infected pest I has a latent period, the period between the instant of infection and that of lysis, during which the virus is reproduced inside the larval tissue. The lysis death rate ξ gives a measure of such latency period T , given by T=1/ξ . The equation of infected class I takes the form [figure omitted; refer to PDF] The lysis largely produces virus polyhedra or polyhedral inclusion bodies (PIB) on average k PIB per insect; k (k>1) is called the virus replication parameter. The virus particles V have natural mortality denoted by _μv due to temperature changes, enzymatic attack, pH dependence, and so forth. The equation of the virus is given by [figure omitted; refer to PDF] We assume the natural enemy or predator P consumes only the infected prey I , the consuming means is the standard bilinear expression lIP , and l is the predation coefficient. We assume the natural enemies in the system remain unharmed from the virus. Then the equations of the infected pest and the natural enemy are [figure omitted; refer to PDF] where d is the death rate of the predator, _...P is the density restriction coefficient of predator, and C is the conversion factor for the predator. Keeping these in view, the dynamics of the epidemic-species may be governed by the following autonomous system of differential equations: [figure omitted; refer to PDF] In the above model, all of the coefficients, r , K , λ , l , ξ , d , _...P , C , _μv , and k , are positive real numbers.

If using the dimensionless time τ=λKt and the transformations s=S/K , i=I/K , p=lP/K , and v=V/K , we have the dimensionless form of model (6). For convenience, we still write t in place of τ ; the corresponding dimensionless form of model (6) is given as follows: [figure omitted; refer to PDF] where we denote a=r/(λK) , η=ξ/(λK) , ...=_...P /(λl) , c=(Cl)/λ , and μ=_μv /(λK) .

3. Equilibria of System (7)

Equilibria of model (7) are obtained by solving ds/dt=di/dt=dp/dt=dv/dt=0 . It can be checked that model (7) has the following three boundary equilibria: the vanishing equilibrium _E0 (0,0,0,0) ; the disease-free equilibrium _E1 (1,0,0,0) ; and the predator-free equilibrium _E2 (s¯,i¯,0,v¯) , where s¯=μ/k , i¯=aμ(k-μ)/k(kη+aμ) , and v¯=aη(k-μ)/(kη+aμ) ; when k>μ , the equilibrium _E2 is nonnegative. The positive equilibrium ^E* (^s* ,^i* ,^p* ,^v* ) is obtained by solving the following equations: [figure omitted; refer to PDF] From the fourth equation of the above equations, we have ^v* =(kη/μ)^i* ; substituting it into the first equation, then ^s* =1-(1+kη/aμ)^i* ; from the third equation of (8), ^p* =(-d+c^i* )/... ; substituting ^s* , ^v* , and ^p* into the second equation, we know the infected pest satisfies the following equation: -(kη/μ+^k2η2 /(a^μ2 )+c/(λ...))^{(i* )2} +(kη/μ+d/(λ...)-η)^i* =0 , so ^i* =(ηaμλ...(k-μ)+ad^μ2 )/(kηaμλ...+^k2η2 λ...+ac^μ2 ) . For the positivity of ^E* , we assume k>μ and d/c<(ηaμλ...(k-μ)+ad^μ2 )/(kηaμλ...+^k2η2 λ...+ac^μ2 )<aμ/(aμ+kη) . If k<μ , then the predator-free equilibrium _E2 and the positive equilibrium ^E* do not exist, which implies the virus replication parameter k is too small to support the virus invasion in insect pest. When the virus replication parameter k is larger than μ and very close to μ , then the predator-free equilibrium _E2 will collapse to the disease-free equilibrium _E1 . Thus it can be seen that the virus replication parameter k plays an important role for the dynamics of system (7). In summary, we have the following theorem about the equilibria of system (7).

Theorem 1.

System (7) always has two equilibria, namely, the vanishing equilibrium _E0 (0,0,0,0) and the disease-free equilibrium _E1 (1,0,0,0) ; if k>μ , there is one more predator-free equilibrium _E2 (s¯,i¯,0,v¯) ; further, if the virus replication parameter satisfies k>μ and d/c<(ηaμλ...(k-μ)+ad^μ2 )/(kηaμλ...+^k2η2 λ...+ac^μ2 )<aμ/(aμ+kη) , then the positive equilibrium ^E* exists.

4. Local Stability and Hopf Bifurcation

The dynamical behaviour of equilibria can be studied by computing variational matrices corresponding to each equilibrium. The variational matrix of (7) is [figure omitted; refer to PDF]

For the vanishing equilibrium _E0 , correspondingly, variational matrix is [figure omitted; refer to PDF] Obviously, _E0 is unstable saddle point with three-dimensional stable manifold and one-dimensional unstable manifold.

For the disease-free equilibrium _E1 , the corresponding variational matrix is [figure omitted; refer to PDF] The characteristic equation is (Λ+a)(Λ+d)[(Λ+η)(Λ+μ)-kη]=0 ; that is, (Λ+a)(Λ+d)[^Λ2 +(μ+η)Λ-η(k-μ)]=0 . When 1<k<μ , there are four real negative eigenvalues and consequently the disease-free equilibrium _E1 is locally asymptotically stable; when k=μ , corresponding to the disease-free equilibrium _E1 , there are a simple eigenvalue 0 and another three negative eigenvalues. The system (7) enters into a saddle-node bifurcation at _E1 ; when k>μ , there are three real negative eigenvalues and one positive eigenvalue so the disease-free equilibrium _E1 is unstable.

For the predator-free equilibrium _E2 , the corresponding variational matrix is [figure omitted; refer to PDF] The characteristic equation is (Λ+d-ci¯)(^Λ3 +_d1^Λ2 +_d2 Λ+_d3 )=0 , where _d1 =μ+η+as¯ , _d2 =as¯(μ+η+v¯) , and _d3 =aημ(1-s¯) . The characteristic equation has clearly one negative real root, namely, ci¯-d , and another three roots are given by the equation ^Λ3 +_d1^Λ2 +_d2 Λ+_d3 =0 . _d1 , _d2 , and _d3 can be looked as the function of s¯ ; since s¯=μ/k and the predator-free equilibrium _E2 exists if k>μ , s¯∈(0,1) . Now we apply Routh-Hurwitz criteria and the definition of a simple criterion of Hopf bifurcation [20] to show the local stability of _E2 and the Hopf bifurcation at _E2 . That is to say, for any s¯∈(0,1) , _E2 is locally asymptotically stable if and only if _di (s¯)>0 , i=1,2,3 , and H(s¯)=_d1 (s¯)_d2 (s¯)-_d3 (s¯)>0 . A Hopf bifurcation of the predator-free equilibrium _E2 occurs at s¯=_s¯0 ∈(0,1) if and only if H(_s¯0 )=0 , _{(dH/ds¯)|s¯0} ...0;0 .

In fact, _di (s¯)>0 , i=1,2,3 , is obvious under the condition k>μ . For the function H(s¯) , s¯∈(0,1) , since H(0)=-aημ<0 and H(1)=(μ+η+a)a(μ+η+v¯)>0 , based on the intermediate value theorem, there exists a _s¯0 ∈(0,1) so that H(_s¯0 )=0 . In addition, dH/ds¯=a_d2 (s¯)+_d1 (s¯)a(μ+η+v¯)+aημ>0 and dH/ds¯>0 at s¯=0 ; at the same time ^d2 H/d^s¯2 =2^a2 (μ+η+v¯)>0 , so _{(dH/ds¯)|s¯0} >0 . Thus H(s¯)>0 at s¯∈(_s¯0 ,1) which implies that the predator-free equilibrium _E2 is locally asymptotically stable in (_s¯0 ,1) and unstable in (0,_s¯0 ) ; further a Hopf bifurcation of the predator-free equilibrium _E2 occurs at s¯=_s¯0 ∈(0,1) ; for decreasing s¯∈(0,_s¯0 ) , it approaches a periodic solution, whose period is 2π/_ω0 , where _ω0 =_d2 (_s¯0 )=_d3 (_s¯0 )/_d1 (_s¯0 ) .

For the positive equilibrium ^E* , the corresponding variational matrix is [figure omitted; refer to PDF] For the local stability of positive equilibrium ^E* , we have the following theorem.

Theorem 2.

The positive equilibrium ^E* is locally asymptotically stable if k<(μ/η^s* )(η+^p* /λ) ; the stable manifold ^WS is ^WS ={(s,i,p,v)∈^R4+ |"(s<^s* ,i<^i* ,p>^p* ,v>^v* )∪(s>^s* ,i>^i* ,p<^p* ,v<^v* )} .

Proof.

We choose the Lyapunov function as follows: [figure omitted; refer to PDF] where _ci >0 , i=1,2,3,4 , is to be determined suitably. Obviously V is positive definite. The derivative of V along the solution of the equation X (t)=_J^E* X(t) , where X(t)=(s(t),i(t),p(t),v(t)^)T , is as follows: [figure omitted; refer to PDF] The symmetric matrix corresponding to V is the following: [figure omitted; refer to PDF] The positive equilibrium ^E* is locally asymptotically stable if V is negative definite, which in turn follows if the symmetric matrix B is negative definite; that is, the odd rank principal minor in order is negative and the even rank principal minor in order is positive, which in turn follows if

(i) -2a_c1^s* <0 ;

(ii) 4a_c1c2 ((1/λ)^p* +η)^s* -(_c2^v* -a_c1^s*)2 >0 ;

(iii): -2a_c1^s* [4_c2c3 ...(η+(1/λ)^p* )^p* -(_c3^p* -(_c2 /λ)^i*)2 ]+2_c3 ...^p* (_c2^v* -a_c1^s*)2 <0 ;

(iv) -2_c4 μ [the left expression of the inequality (iii)]-4a_c1c3 ...^p*s* (_c4 kη-_c2^s*)2 -16a_c1c2c3c4 ...kη(^s*)2p* >0 .

We choose _c2^v* -a_c1^s* =0 , _c3^p* -(_c2 /λ)^i* =0 , and _c4 kη-_c2^s* =0 ; then from the above expression, we know if 16_c4 μa_c1^s*_c2c3 ...(η+(1/λ)^p* )^p* >16a_c1c2c3c4 ...η(^s*)2p* , that is, k<(μ/η^s* )(η+^p* /λ) , for the symmetric matrix B , the odd rank principal minor in order is negative and the even rank principal minor in order is positive. That is to say, if k<(μ/η^s* )(η+^p* /λ) , the positive equilibrium ^E* is locally asymptotically stable. Furthermore, to find out the basin of attraction, we choose the Lyapunov function as follows: [figure omitted; refer to PDF] Differentiating V along the solution of the model and choosing _c1 =a/^v* , we can obtain the stable manifold ^WS ={(s,i,p,v)∈^R4+ |"(s<^s* ,i<^i* ,p>^p* ,v>^v* )∪(s>^s* ,i>^i* ,p<^p* ,v<^v* )} . The proof is completed.

Now, we will find out the conditions for which the equilibrium ^E* enters into Hopf bifurcation. The characteristic equation for ^E* is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since _p2 , _p3 , and _p4 can be written as _p2 =a^s* (...^p* +μ)+...^p* μ+(η+(1/λ)^p* )(a^s* +...^p* )+[μ(η+(1/λ)^p* )-kη^s* ]+a^s*v* +(c/λ)^p*i* , _p3 =a^s* ...^p* μ+(a^s* +...^p* )[μ(η+(1/λ)^p* )-kη^s* ]+(η+(1/λ)^p* )...^p* a^s* +(c/λ)^p*i* (a^s* +μ) +a...^s*p*v* +^v*s* (aμ+kη) , and _p4 =a...^s*p* [μ(η+(1/λ)^p* )-ηk^s* ]+aμ^s* (c/λ)^p*i* +...^s*p*v* (aμ+kη) , respectively, under the condition k<(μ/η^s* )(η+(1/λ)^p* ) , it is not easy to know _pi >0 , i=1,2,3,4 . Let Ψ(k):(0,+∞)[arrow right]R be the following continuously differentiable function: Ψ(k)=_p1 (k)_p2 (k)_p3 (k)-^p32 (k)-_p4 (k)^p12 (k) . A Hopf bifurcation of the equilibrium ^E* occurs at ^k* ∈(1,+∞) if and only if (i) Ψ(^k* )=0 , _{(dReΛ(k)/dk)|^k*} ...0;0 , where Λ(k) is purely imaginary at k=^k* , and (ii) all other eigenvalues are of negative real parts.

The existence of ^k* can be obtained by solving Ψ(k)=0 . At k=^k* , _p1p2p3 =^p32 +_p4^p12 ; that is, _p2 =_p3 /_p1 +_p4p1 /_p3 , which deduces that the characteristic equation can be written as (^Λ2 +_p3 /_p1 )(^Λ2 +_p1 Λ+_p1p4 /_p3 )=0 . So there exist a pair of purely imaginary _Λ1 ,_Λ2 =±_p3 /_p1 , respectively; other eigenvalues _Λ3 , _Λ4 satisfy _Λ3 +_Λ4 =-_p1 , _Λ3Λ4 =_p1p4 /_p3 . Denote _ω0 =Im..._Λ1 (^k* )=_p3 /_p1 ; then ^ω02 +_Λ3Λ4 =(^p32 +^p12_p4 )/_p1p3 =_p2 since Ψ(^k* )=0 , ^ω02 (_Λ3 +_Λ4 )=-_p3 , and ^ω02_Λ3Λ4 =_p4 . If _Λ3 and _Λ4 are complex conjugates, then from _Λ3 +_Λ4 =-_p1 we know Re_Λ3 =-_p1 /2<0 ; if _Λ3 and _Λ4 are real roots, then _Λ3Λ4 >0 based on the fact that ^ω02_Λ3Λ4 =_p4 >0 ; furthermore, since _Λ3 +_Λ4 =-_p1 <0 , then _Λ3 <0 , _Λ4 <0 . In summary, for the characteristic equation, besides _Λ1 and _Λ2 , other eigenvalues are negative real parts. Now we show _{(dReΛ(k)/dk)|^k*} ...0;0 . In fact, let Λ(k)=_β1 (k)+i_β2 (k) ; substituting it into the characteristic equation for the positive equilibrium ^E* and calculating the derivative on k , the following expression can be given through some calculations and arrangement: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It deduces that [figure omitted; refer to PDF] Solving the above equations on ^{β1[variant prime]} , ^{β2[variant prime]} , we have [figure omitted; refer to PDF] At this point, the conditions of Hopf bifurcation for positive equilibrium ^E* are verified. In summary, we have the following theorem.

Theorem 3.

There exists a ^k* ∈(μ,+∞) ; the Hopf bifurcation for the positive equilibrium ^E* occurs at k=^k* . The period of the bifurcating periodic orbits close to k=^k* is 2π/_ω0 , where _ω0 =_p3 (^k* )/_p1 (^k* ) .

In the next section we will consider the stability of bifurcating periodic orbits applying Poore's condition [19].

5. Stability of the Bifurcating Periodic Solution

We apply Poore's condition for verification of the orbital stability of the Hopf bifurcating limit cycle. For convenience, we list the preliminaries firstly. Let a real, n -dimensional (n...5;2 ), first-order system of autonomous differential equations be of the form [figure omitted; refer to PDF] where ν=(_ν1 ,_ν2 ,...,_νm^)T denotes a vector of m -real parameters. There must firstly exist a combination of the parameters, say, _ν0 ; a critical point ^a0 , such that the variational matrix _F^x (^a0 ,^ν0 ) has exactly two, nonzero, purely imaginary eigenvalues, say, ±i_ω0 with _ω0 >0 ; and other n-2 eigenvalues with nonzero real parts. To vary one, some, or all of the m parameters, an m -dimensional vector function f([varepsilon]) is introduced with the property that f(0)=0 , and hence we confine our analysis to the following system of ODE: [figure omitted; refer to PDF] It follows from the definition of F(x,[varepsilon]) that F(^a0 ,0)=0 and eigenvalues of _Fx (^a0 ,0)=0 are the same as those of F^(^a0 ,^ν0 ) . It is assumed that F(x,[varepsilon])∈^Ck [D×(-_{[straight epsilon]0} ,_{[straight epsilon]0} )] , where k...5;3 and D is a domain in ^Rn containing ^a0 and _{[straight epsilon]0} >0 . As there is no nonzero eigenvalue of the variational matrix, _Fx (^a0 ,0)...0;0∈^Rn , and hence the implicit function theorem guarantees the existence of a critical point ^{a[varepsilon]} which is k -times continuously differentiable in [varepsilon] and satisfies F(^{a[varepsilon]} ,[varepsilon])=0 for [varepsilon] in a small neighborhood of [varepsilon]=0 . Using this definition of ^{a[varepsilon]} , a change of variables is introduced: [figure omitted; refer to PDF] This reduces the differential (25) to the following form: [figure omitted; refer to PDF] where ^A0 =_Fx (^a0 ,0) and G is defined as given in [19]. Thus, the problem of periodic solutions of (25) is reduced to a perturbation problem in the small parameter of ^{μ[variant prime]} .

Now, the stability information of the bifurcating periodic orbits is contained in the following theorem of Poore, coupled with an algebraic expression, which completely reduces the determination of stability to an algebraic problem. By the assumptions of Theorem 2.3 [19], the differential equation in (27) is continuously differentiable in ^{μ[variant prime]} , where δ=δ(^{μ[variant prime]} ) and η=η(^{μ[variant prime]} ) , and in the function y in a neighborhood of the periodic orbit. Thus, the existing periodic orbit will be asymptotically orbitally stable with asymptotic phase if n-1 of the characteristic multipliers of the variational equation have moduli less than one. The following theorem developed by Poore about the modulus of each of the characteristic multipliers reduces the condition of orbital stability of the Hopf bifurcating limit cycle to an algebraic expression as follows.

Theorem 4.

Let F(x,[varepsilon]) satisfy the hypotheses in Theorem 2.3 [19] and let u and v denote the left and right eigenvectors, respectively, for the eigenvalues i_ω0 (_ω0 >0) of the variational matrix ^A0 . If u and v are normalized in the sense that uv=1 , then [figure omitted; refer to PDF] where ^A0 =_Fx (^a0 ,0) , _Fxx =_Fxx (^a0 ,0) , _Fxxx =_Fxxx (^a0 ,0) , b(0) is the ^{μ[variant prime]} =0 value of the two-dimensional vector b(^{μ[variant prime]} ) , which occurs in ...y(s,^{μ[variant prime]} )=Φ(s)b(^{μ[variant prime]} ) , and v¯ denotes the complex conjugate of v .

Written out in component form, the above expression reduces to [figure omitted; refer to PDF] where the repeated indices within each term imply a sum from 1 to n and all the derivatives of F are evaluated at the equilibrium ^a0 . The sign of the real and imaginary parts of the right-hand side of expression in Theorem 4 is independent of the choice of b(^{μ[variant prime]} ) in ...y(s,^{μ[variant prime]} )=Φ(s)b(^{μ[variant prime]} ) and the eigenvectors u and v so long as b(0)·b(0)>0 and uv=1 . So positivity of the real part of the above expression in parenthesis really indicates the orbital stability of the periodic solution arising out of Hopf bifurcation.

For model (7) we considered, let the right four equations be ^F1 =as(1-s-i)-vs , ^F2 =vs-(1/λ)ip-ηi , ^F3 =p(-d-...p+ci) , and ^F4 =-μv+kηi , respectively. To verify that the conditions of Theorem 4 are satisfied, we will calculate the second- and the third-order derivatives of ^Fl , l=1,2,3,4 , as follows: [figure omitted; refer to PDF] Other second-order and third-order derivatives are all zero. The inverse matrix Q of _J^E* -2i_ω0 and the left and right eigenvectors u and v , respectively, in the sense that uv=1 , of the variational matrix _J^E* can be calculated. Based on the preliminary work, we can calculate expression (29) in parenthesis as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Putting the values of _Fxxx , _Fxx , u , and v and components of the matrix Q in terms of the parameters of the model, the positivity of the real part of expression (29) in parenthesis can be deduced. This in turn indicates the orbital stability of the limit cycle arising out of Hopf bifurcation.

6. Extinction and Permanence with Impulsive Effect

In the sections ahead, we focus our attention on model (6) or (7), in which the growth of the predator and the virus is continuous. But in fact, it is needed to throw in the predator or virus according to the pest control at some moments for better effects. So in this section, we develop model (6) into an impulsive system as follows and the presence of impulses gives the system a mixed nature, both continuous and discrete: [figure omitted; refer to PDF] where the predator and virus are thrown at nτ , n=0,1,2,... , τ is the period, and b , q are the amount of impulsive stocking on the predator and the virus; b , q are positive constants. Other parameters are the same as model (6). S(n^τ+ ) , I(n^τ+ ) , P(n^τ+ ) , and V(n^τ+ ) represent the densities of the susceptible pest, the infected pest by virus, the predator, and the virus, respectively: S(n^τ+ )=_{lim...t[arrow right]n^τ+} S(t) , I(n^τ+ )=_{lim...t[arrow right]n^τ+} I(t) , P(n^τ+ )=_{lim...t[arrow right]n^τ+} P(t) , and V(n^τ+ )=_{lim...t[arrow right]n^τ+} V(t) ; S(t) , I(t) , P(t) , and V(t) are left continuous at nτ ; that is, S(nτ)=_{lim...t[arrow right]n^τ-} S(t) , I(nτ)=_{lim...t[arrow right]n^τ-} I(t) , P(nτ)=_{lim...t[arrow right]n^τ-} P(t) , and V(nτ)=_{lim...t[arrow right]n^τ-} V(t) . In this section, we mainly consider the global stability of pest-eradication solution and the uniform permanence of system (33); before introducing our main results, we give some preliminaries needed next.

For the system [figure omitted; refer to PDF] we have the following lemma.

Lemma 5.

System (34) has a unique positive periodic solution ^V* (t) with period τ if and only if the initial value ^V* (⁰⁺ )=q/(1-^{e-_μv τ} ) , where ^V* (t)=q^{e-_μv (t-nτ)} /(1-^{e-_μv τ} ) , nτ<t...4;(n+1)τ , and ^V* (t) is globally asymptotically stable; in addition, every solution V(t) of system (34) can be written as V(t)=^V* (t)+(V(⁰⁺ )-^V* (⁰⁺ ))^{e-_μv t} , nτ<t...4;(n+1)τ .

For the system [figure omitted; refer to PDF] we have the following lemma.

Lemma 6.

System (35) has a unique positive periodic solution ^P* (t) with period τ if and only if the initial value ^P* (⁰⁺ )=-[(1-^edτ )(d/_...p -b)+Δ]/[2(1-^edτ )] , where Δ=(1-^edτ)2 (d/_...p -b⁾² -(4d/_...p )(1-^edτ )b^edτ , ^P* (t)=(-d/_...p )/(1-(d/^P*_...p +1)^ed(t-nτ) ) , nτ<t...4;(n+1)τ , and ^P* (t) is globally asymptotically stable.

Lemma 7.

There exists a constant L>0 such that S(t) , I(t) , P(t) , and V(t)...4;L for each solution of (33) with t being large enough.

The solution of system (33) corresponding to I=0 is called pest-eradication solution (0,0,^P* (t),^V* (t)) whose explicit form can be obtained by Lemmas 5 and 6, where ^P* (t)=(-d/_...p )/(1-(d/^P*_...p +1)^ed(t-nτ) ) and ^V* (t)=q^{e-_μv (t-nτ)} /(1-^{e-_μv τ} ) , nτ<t...4;(n+1)τ , with initial values ^P* (⁰⁺ ) , ^V* (⁰⁺ ) as Lemmas 5 and 6, respectively. Now we consider the stability of pest-eradication solution. The Jacobi matrix A(t) at (0,0,^P* (t),^V* (t)) is [figure omitted; refer to PDF] we can calculate the monodromy matrix M(T)=B^{e∫0T ...A(t)dt} , where B is identity matrix and M(T) is just as follows: [figure omitted; refer to PDF] there is no need to calculate the exact forms of Δ since they are not required for the following analyses. The Floquet multipliers are _μ1 =exp...(rτ-λ^∫0τ ...^V* dt) , _μ2 =exp...(-l^∫0τ ...^P* dt-ξτ) , _μ3 =exp...(-dτ-2_...p^∫0τ ...^P* dt) , and _μ4 =^{e-_μv τ} , respectively. Obviously, 0<_μ2 ,_μ3 ,_μ4 <1 , so the stability of (0,0,^P* (t),^V* (t)) is decided by whether _μ1 =exp...(rτ-λ^∫0τ ...^V* dt)<1 ; that is, when τ<λq/r_μv , the pest-eradication solution is locally stable; further we can prove that it is globally asymptotically stable.

Theorem 8.

If τ<λq/r_μv , then the pest-eradication solution (0,0,^P* (t),^V* (t)) of system (33) is globally asymptotically stable.

The pest-eradication solution (0,0,^P* (t),^V* (t)) is globally asymptotically stable when τ<λq/r_μv ; that is to say, the pest population is eradicated totally. But in practice, we only need to control the pest population under the economic threshold level (ETL), instead of eradicating it totally, and hope the pest population, the predator (natural enemy), and the virus population can coexist when the pests do not bring about immense economic losses. As for this, we have the uniform permanence of system (33); see the next theorem.

Theorem 9.

If τ>λq/r_μv , then system (33) is uniformly permanent.

The proofs of the lemmas and the theorems listed in this section are all omitted here since they are similar to Hui and Zhu [21].

7. Numerical Simulations

In this section, we use numerical simulations to illustrate the dynamical behaviors of system (7) and (33). We choose k as bifurcation parameter; for system (7), the other parameter values are taken as a=0.1 , λ=15 , η=0.06 , d=0.1 , ...=0.05 , c=0.5 , and μ=0.6 ; for system (33), the other parameter values are taken as r=20 , K=10000 , λ=0.001 , l=0.01 , ξ=10 , d=3.1 , _...p =0.005 , C=0.5 , _μv =10 , b=1 , q=2 , and τ=1 .

Firstly, Figure 1 illustrates that the disease-free equilibrium _E1 is locally asymptotically stable when k...4;μ . It is calculated that _di (s¯)>0 (i=1,2,3 ) and H(s¯)>0 with k∈(0.6,8.865496710) . Therefore, the disease-free equilibrium _E1 is unstable and _E2 is locally asymptotically stable when k∈(0.6,8.865496710) (see Figure 2). A Hopf bifurcation of the predator-free equilibrium _E2 occurs at k=8.865496710 due to H(s¯)=0 and _{(dH/ds¯)|s¯0} ...0;0 . In this case, there is a periodic solution for system (7) (see Figure 3). When k∈(1,3) , the positive equilibrium ^E* exists and it is locally asymptotically stable. Figure 4 illustrates the stability of the positive equilibrium ^E* .

Figure 1: Locally asymptotical stability of the disease-free equilibrium _E1 when k...4;μ , where k=0.6 .

[figure omitted; refer to PDF]

Figure 2: Locally asymptotical stability of the predator-free equilibrium _E2 when k>μ and H(s¯)>0 , where k=4 .

[figure omitted; refer to PDF]

Figure 3: A Hopf bifurcation of the predator-free equilibrium _E2 occurs when k>μ and H(s¯)=0 , _{(dH/ds¯)|s¯0} ...0;0 , where k=9 .

[figure omitted; refer to PDF]

Figure 4: Stability of the positive equilibrium ^E* when k<μ(η+^p* /λ)/(η^s* ) , where k=2 .

[figure omitted; refer to PDF]

Figure 5 is the bifurcation graph of system (7). The virus replication factor k is chosen as the bifurcation parameter. When 1<k<8.866 , there is a stable positive equilibrium ^E* ; when k>8.866 , the positive equilibrium ^E* is unstable and the Hopf bifurcation appears at k=8.86 ; the stable periodic solution appears. Figure 6 illustrates the bifurcation phenomenon of the impulsive system (33). When 1<k<14 , there exists a stable periodic solution; when k>14 , the system (33) enters into the chaos state.

Figure 5: Bifurcation of the system (7): bifurcation parameter is chosen as the virus replication factor k , where other parameters are a=0.1 , λ=15 , η=0.06 , d=0.1 , ...=0.05 , c=0.5 , and μ=0.6 . Horizontal axis represents the bifurcation parameter k and vertical axis represents the positive equilibrium state of the susceptible pest; the full line represents stable equilibrium state and the dotted line represents unstable equilibrium state; the small dots represent the maximum and the minimum amplitude of limit cycle bifurcating from the positive equilibrium and the Hopf bifurcation appears at k=8.86 .

[figure omitted; refer to PDF]

Figure 6: Bifurcation of the impulsive system (33): bifurcation parameter is chosen as the virus replication factor k , where other parameters are r=20 , K=10000 , λ=0.001 , l=0.01 , ξ=10 , d=3.1 , _...p =0.005 , C=0.5 , _μv =10 , b=1 , q=2 , and τ=1 . Horizontal axis represents the bifurcation parameter k and vertical axis represents the densities of the infected pest.

[figure omitted; refer to PDF]

8. Conclusions

In this paper, from Sections 1 to 5 we focus our attention on an epidemic-species hybrid system without impulse influence. From our analyses, we found the virus replication factor k plays a very important role for dynamics of system. The system always has two equilibria, namely, the vanishing equilibrium _E0 (0,0,0,0) and the disease-free equilibrium _E1 (1,0,0,0) . When the virus replication factor k is larger than its natural death rate μ , that is, k>μ , the predator-free equilibrium appears; furthermore, the virus replication factor k satisfies k>μ and d/c<(ηaμλ...(k-μ)+ad^μ2 )/(kηaμλ...+^k2η2 λ...+ac^μ2 )<aμ/(aμ+kη) ; the positive equilibrium ^E* exists. The vanishing equilibrium _E0 (0,0,0,0) is a saddle point with three-dimensional stable and one-dimensional unstable manifold. Biologically, it means that in the absence of prey, that is, the insect pest, the predator and virus population becomes extinct. When 1<k<μ , the disease-free equilibrium _E1 is locally asymptotically stable; it means that if the virus replication parameter or virulence of the applied virus on the system is less than its natural death rate μ , then only k viruses obtained from lysis of the infected pest are not enough to maintain the viral infection in the pest population. When k=μ , the system passes through saddle-node bifurcation and for value of k higher than μ , the system gives two branches of equilibria, namely, the disease-free equilibrium _E1 and the predator-free equilibrium _E2 , one hyperbolic equilibrium with three-dimensional stable manifold and one nonhyperbolic equilibrium, respectively; that is, for k>μ , _E1 becomes unstable and _E2 becomes feasible. Biologically, for spreading of infection into the system, the virus replication factor k must be greater than μ . When k>μ , the predator-free equilibrium _E2 is feasible and stable. For higher value of k , there exists a _s¯0 which deduces that there exists a _k0 such that at k=_k0 the system enters into a Hopf bifurcation. When k>_k0 the solution becomes unstable and gives a periodic orbit of small amplitude near _k0 . For the positive equilibrium ^E* , there exists a ^k* ∈(μ,+∞) , the positive equilibrium is locally asymptotically stable if μ<k<^k* , and the Hopf bifurcation occurs at k=^k* , bifurcating a periodic orbit which is orbitally stable applying Poore's condition for verification. Considering, in practice, that the predator and the virus are thrown at some moments, lastly in Section 6, we think about the corresponding impulsive equation to model (6). The results have shown that when the impulsive releasing period τ<λq/r_μv , then the pest-eradication solution (0,0,^P* (t),^V* (t)) of system is globally asymptotically stable; otherwise the system is uniformly permanent. So the pest management can be realized by controlling the impulsive releasing period.

Acknowledgments

The authors would like to thank the referee, Professor H. F. Huo, for his careful reading of the original paper and giving meaningful suggestions to improve the quality of the paper. At the same time, they thank Professor S. Y. Tang for his instructive advice and useful suggestions for the numerical simulation. This research was supported by Guangxi Natural Science Foundation 2012GXNSFAA053011 and Foundation of Guangxi Educational Committee (no. YB2014203).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] B. S. Goh Management and Analysis of Biological Populations , Elsevier Science, Amsterdam, The Netherlands, 1980.

[2] L. A. Lacey, M. S. Goettel, "Current developments in microbial control of insect pests and prospects for the early 21st century," Entomophaga , vol. 40, no. 1, pp. 3-27, 1995.

[3] D. R. Dent, H. F. Evans, "Integrated pest management and microbial insecticides," Microbial Insecticides: Novelty or Necessity? , of Proceedings of the British Crop Protection Council Symposium, 68, pp. 127-138, 1997.

[4] R. Georgis, H. F. Evans, "Commercial prospects of microbial insecticides in agriculture," Microbial Insecticides: Novelty or Necessity? , vol. 68, of Proceedings of the British Crop Protection Council Symposium, pp. 243-252, 1997.

[5] G. M. Tatchell, H. F. Evans, "Microbial insecticides and IPM: current and future opportunities for the use of biopesticides," Microbial Insecticides: Novelty or Necessity? , of Proceedings of the British Crop Protection Council Symposium, 68, pp. 191-200, 1997.

[6] H. F. Evans, H. F. Evans, "The role of microbial insecticides in forest pest management," Microbial Insecticides: Novelty or Necessity? , vol. 68, of Proceedings of the British Crop Protection Council Symposium, pp. 29-40, 1997.

[7] K. van Frankenhuyzen, R. C. Reardon, N. R. Debois, L. A. Lacey, H. K. Kaya, "Forest defoliators," Field Manual of Techniques in Invertebrate Pathology: Application and Evaluation of Pathogens for Control of Insects and Other Invertebrate Pests , pp. 767-785, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[8] O. Skovmand, J. Kerwin, L. A. Lacey, L. A. Lacey, H. K. Kaya, "Microbial control of mosquitoes and black flies," Field Manual of Techniques in Invertebrate Pathology , pp. 767-785, Kluwer Academic, Dordrecht, The Netherlands, 2000.

[9] F. Fenner, F. N. Ratcliff Myxomatosis , Cambridge University Press, Cambridge, UK, 1965.

[10] H. D. Burges, N. W. Hussey Microbial Control of Insects and Mites , Academic Press, New York, NY, USA, 1971.

[11] R. M. Anderson, R. M. May, "Regulation and stability of host-Cparasite intercations. I. Regulatory Processes," Journal of Animal Ecology , vol. 47, pp. C219-C247, 1978.

[12] B. S. Goh, J. Leitman, T. L. Vincent, B. S. Goh, "Optimal epidemic programs for pest control," Management and Analysis of Biological Populations , 1980.

[13] P. Waltman, S. A. Levin, S. A. Levin Deterministic Threshold Models in the Theory of Epidemics , of Lecture Notes in Bioinformatics, Springer, Berlin, Germany, 1974.

[14] N. T. J. Bailey The Mathematical Theory of Infectious Diseases and its Application , Griffin, London, UK, 1975.

[15] K. Wickwire, "Mathematical models for the control of pests and infectious diseases: a survey," Theoretical Population Biology , vol. 11, no. 2, pp. 182-238, 1977.

[16] H. R. Thieme Mathematics in Population Biololgy , Princeton University Press, Princeton, NJ, USA, 2003.

[17] E. Kurstak Microbial and Viral Pesticide , Marcel and Dekker, New York, NY, USA, 1982.

[18] M. B. Thomas, S. Blandford, C. Gbongboui, C. J. Lomer, "Experimental studies to evaluate spray applications of a mycoinsecticide against the rice grasshopper, Hieroglyphus daganensis, in northern Benin," Entomologia Experimentalis et Applicata , vol. 87, no. 1, pp. 93-102, 1998.

[19] A. B. Poore, "On the theory and application of the Hopf-Friedrichs bifurcation theory," Archive for Rational Mechanics and Analysis , vol. 60, no. 4, pp. 371-393, 1976.

[20] J. Guckenheimer, P. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , vol. 42, of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.

[21] J. Hui, D. Zhu, "Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects," Chaos, Solitons and Fractals , vol. 29, no. 1, pp. 233-251, 2006.