(ProQuest: ... denotes non-US-ASCII text omitted.)
Lingshu Wang 1 and Guanghui Feng 2
Academic Editor:Shan Zhao
1, School of Mathematics and Statistics, Hebei University of Economics & Business, Shijiazhuang 050061, China
2, Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
Received 24 November 2013; Revised 11 February 2014; Accepted 12 February 2014; 23 March 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
Since the pioneering work of Kermack-Mckendrick on SIRS [1], epidemiological models have received much attention from scientists. Mathematical models have become important tools in analyzing the spread and control of infectious disease. It is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. Ecoepidemiology which is a relatively new branch of study in theoretical biology, tackles such situations by dealing with both ecological and epidemiological issues. It can be viewed as the coupling of an ecological predator-prey model and an epidemiological SI, SIS, or SIRS model. Following Anderso and May [2] who were the first to propose an ecoepidemiological model by merging the ecological predator-prey model introduced by Lotka and Volterra, the effect of disease in ecological system is an important issue from mathematical and ecological point of view. Many works have been devoted to the study of the effects of a disease on a predator-prey system [1-5]. In [5], Xiao and Chen have considered a ratio-dependent predator-prey system with disease in the prey. Consider [figure omitted; refer to PDF] where S(t) and I(t) represent the densities of susceptible and infected prey population at time t , respectively, and Y(t) represents the density of the predator population at time t . The parameters r , K , β , d , b , a , p , and c are positive constants representing the prey intrinsic growth rate, carrying capacity, transmission rate, the infected prey death rate, capturing rate, half capturing saturation constant, conversion rate, and the predator death rate, respectively. A periodic solution can occur whether the system (1) is permanent or not; that is, there are solutions which tend to disease-free equilibrium while bifurcating periodic solution exists.
Recently, the qualitative analysis of predator-prey models incorporating a prey refuge has been done by many authors, see [3, 4]. In [3], Pal and Samanta incorporated a prey refuge (1-m)I into system (1). Sufficient conditions were derived for the stability of the equilibria of the system.
We note that it is assumed in system (1) that each individual predator admits the same ability to feed on prey. This assumption seems not to be realistic for many animals. In the natural world, there are many species whose individuals pass through an immature stage during which they are raised by their parents, and the rate at which they attack prey can be ignored. Moreover, it can be assumed that their reproductive rate during this stage is zero. Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. Stage-structured models have received great attention in recent years (see, e.g., [6-9]).
Time delays of one type or another have been incorporated into biological models by many researchers (see, e.g., [8-11]). In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause the population to fluctuate. Time delay due to gestation is a common example, because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays.
Based on the above discussions, in this paper, we incorporate a prey refuge, stage structure for the predator, and time delay due to the gestation of predator into the system (1). To this end, we study the following differential equations: [figure omitted; refer to PDF] where Y1 (t) and Y2 (t) represent the densities of the immature and the mature predator population at time t , respectively, the parameters d1 , d2 , and r1 are positive constants in which d1 and d2 are the death rates of the immature and the mature predator, respectively, r1 denotes the rate of immature predator becoming mature predator, the constant proportion infected prey refuge is (1-m)I , where m∈[0,1) is a constant, and τ...5;0 is a constant delay due to the gestation of the predator.
The initial conditions for system (2) take the form [figure omitted; refer to PDF] where R+04 ={(x1 ,x2 ,x3 ,x4 ):xi ...5;0,i=1,2,3,4} .
It is well known by the fundamental theory of functional differential equations [12] that system (2) has a unique solution (S(t),I(t),Y1 (t),Y2 (t)) satisfying initial conditions (3).
The organization of this paper is as follows. In the next section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3). In Section 3, we investigate the global stability of the predator-extinction equilibrium. In Section 4, we establish the local stability and the global attractivity of the coexistence equilibrium of system (2). Further, we study the existence of Hopf bifurcation for system (2) at the positive equilibrium. A brief discussion is given in Section 5 to conclude this work.
2. Preliminaries
In this section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3).
Theorem 1.
Solutions of system (2) with initial conditions (3) are positive, for all t...5;0 .
Proof.
Let (S(t),I(t),Y1 (t),Y2 (t)) be a solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that [figure omitted; refer to PDF]
Let us consider Y1 (t) and Y2 (t) , for t∈[0,τ] . Since ψ2 (θ)...5;0 for θ∈[-τ,0] , we derive from the third equation of system (2) that [figure omitted; refer to PDF] Since ψ1 (0)>0 , a standard comparison argument shows that [figure omitted; refer to PDF] that is, Y1 (t)>0 for t∈[0,τ] . For t∈[0,τ] , it follows from the fourth equation of (2) that [figure omitted; refer to PDF] Since ψ2 (0)>0 , a standard comparison argument shows that [figure omitted; refer to PDF] that is, Y2 (t)>0 for t∈[0,τ] . In a similar way, we treat the intervals [τ,2τ],...,[nτ,(n+1)τ], n∈N . Thus, S(t)>0 , I(t)>0 , Y1 (t)>0 , and Y2 (t)>0 , for all t...5;0 . This completes the proof.
Theorem 2.
Positive solutions of system (2) with initial conditions (3) are ultimately bounded.
Proof.
Let (S(t),I(t),Y1 (t),Y2 (t)) be any positive solution of system (2) with initial conditions (3). Denote d^=min...{d,d1 ,d2 } . Define [figure omitted; refer to PDF] Calculating the derivative of V(t) along positive solutions of system (2), it follows that [figure omitted; refer to PDF] which yields [figure omitted; refer to PDF] If we choose M1 =(K(r+d^)2 )/4rd^ and M2 =(pK(r+d^)2 )/4rd^ , then [figure omitted; refer to PDF] This completes the proof.
3. Predator-Extinction Equilibrium and Its Stability
In this section, we discuss the stability of the predator-extinction equilibrium.
It is easy to show that if Kβ>d , system (2) admits a predator-extinction equilibrium E1 (S1 ,I1 ,0,0) , where [figure omitted; refer to PDF]
The characteristic equation of system (2) at the equilibrium E1 is of the form [figure omitted; refer to PDF] where g1 =(r1 +d1 +d2 ), g0 =d2 (r1 +d1 ), f0 =-pbr1 . When τ=0 , if d2 (r1 +d1 )>pbr1 , then E1 is locally asymptotically stable and if d2 (r1 +d1 )<pbr1 , then E1 is unstable. It is easily seen that [figure omitted; refer to PDF] Hence, if d2 (r1 +d1 )>pbr1 , by Lemma B in [11], it follows that the equilibrium E1 is locally asymptotically stable for all τ...5;0 . If d2 (r1 +d1 )<pbr1 , then E1 is unstable for all τ...5;0 .
Theorem 3.
Let Kβ>d hold; the predator-extinction equilibrium E1 is globally stable provided that [figure omitted; refer to PDF]
Proof.
Based on above discussions, we only prove the global attractivity of the equilibrium E1 . Let (S(t),I(t),Y1 (t),Y2 (t)) be any positive solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that [figure omitted; refer to PDF] Consider the following auxiliary equations: [figure omitted; refer to PDF] If Kβ>d , then by Theorem 3.1 in [4], it follows from (19) that [figure omitted; refer to PDF] By comparison, we obtain that [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T1 >0 such that if t>T1 , then I(t)...4;I1 +[straight epsilon] .
It follows from the third and the fourth equations of system (2) that, for t>T1 +τ , [figure omitted; refer to PDF] Consider the following auxiliary equations: [figure omitted; refer to PDF] If d2 (r1 +d1 )>r1 pb , then by Lemma 2.4 in [9], it follows from (23) that [figure omitted; refer to PDF] By comparison, we obtain that [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T2 >0 such that if t>T2 , then Y2 (t)...4;[straight epsilon] .
It follows from the first and the second equations of system (2) that for t>T2 : [figure omitted; refer to PDF] Consider the following auxiliary equations: [figure omitted; refer to PDF] If Kβ>d , and (1-m)<(a/b)(Kβ-d) , then by Theorem 3.1 in [4], it follows from (27) that [figure omitted; refer to PDF] By comparison, for [straight epsilon] sufficiently small, we obtain that [figure omitted; refer to PDF] which, together with (21), yields [figure omitted; refer to PDF]
Hence, if Kβ>d, d2 (r1 +d1 )>pbr1 , (1-m)<(a/b)(Kβ-d) hold, then the equilibrium E1 (S1 ,I1 ,0,0) is globally stable.
4. Coexistence Equilibrium and Its Stability
In this section, we discuss the stability of the coexistence equilibrium and the existence of a Hopf bifurcation. It is easy to show that if the following holds:
(H1) d2 (r1 +d1 )<r1 pb , 0<(1-m)<(ar1 p(Kβ-d))/(r1 pb-d2 (r1 +d1 )) ,
then system (2) has a unique coexistence equilibrium E* (S* ,I* ,Y1* ,Y2* ) , where [figure omitted; refer to PDF]
The characteristic equation of system (2) at the equilibrium E* takes the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF]
When τ=0 , (32) becomes [figure omitted; refer to PDF] If the following holds:
(H2) (r/K)S* -α1 >0 , β((r/K)+β)I* -(r/K)α1 >0 ,
then it is easy to show that [figure omitted; refer to PDF] If (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]>p32 (p0 +q0 ) , then, by the Routh-Hurwitz theorem, when τ=0 , the coexistence equilibrium E* of system (2) is locally asymptotically stable and E* is unstable if (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]<p32 (p0 +q0 ) .
If iω(ω>0) is a solution of (34), separating real and imaginary parts, we have [figure omitted; refer to PDF] By squaring and adding the two equations of (36), it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] If h3 >0, h2 >0, h1 >0 and p0 -q0 >0 , by the general theory on characteristic equation of delay differential equation from [13] (Theorem 4.1), E* remains stable for all τ>0 .
If hi >0, (i=1,2,3) and p0 -q0 <0 , then (37) has a unique positive root ω0 ; that is, (34) admits a pair of purely imaginary roots of the form ±ω0 . From (36), we see that [figure omitted; refer to PDF] By Theorem 3.4.1 in [13], we see that E* remains stable for τ<τ0 .
In the following, we claim that [figure omitted; refer to PDF] This will show that there exists at least one eigenvalue with a positive real part for τ>τ0 . Moreover, the conditions for the existence of a Hopf bifurcation (Theorem 2.9.1 in [13]) are then satisfied yielding a periodic solution. To this end, by differentiating equation (34) with respect to τ , it follows that [figure omitted; refer to PDF] Hence, a direct calculation shows that [figure omitted; refer to PDF] We derive from (36) that [figure omitted; refer to PDF] Hence, it follows that [figure omitted; refer to PDF] Therefore, the transversal condition holds and a Hopf bifurcation occurs at ω=ω0 ,τ=τ0 .
In conclusion, we have the following results.
Theorem 4.
For system (2), let (H1) and (H2) hold; we have the following:
(i) if (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]>p32 (p0 +q0 ) , hi >0 , and p0 -q0 >0 , then the coexistence equilibrium E* is locally asymptotically stable, for all τ...5;0 ;
(ii) if (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]>p32 (p0 +q0 ) , hi >0 , and p0 -q0 <0 , then there exists a positive number τ0 , such that the coexistence equilibrium E* is locally asymptotically stable if 0...4;τ<τ0 and is unstable for τ>τ0 ; further, system (2) undergoes a Hopf bifurcation at E* when τ=τn , n=0,1,2,... ;
(iii): if (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]<p32 (p0 +q0 ) , then the coexistence equilibrium E* is unstable, for all τ...5;0 .
We now give some examples to illustrate the main results above.
Example 5.
In (2), we let a=2, p=0.9 , r=10 , r1 =0.5, k=1, β=1 , d=d1 =d2 =0.1 , b=1 , and m=0.5 . System (2), with the above coefficients, has a unique coexistence equilibrium E* (0.3167,0.6212,0.2019,1.0095) . It is easy to show that (r/K)S* -α1 [approximate]3.1468>0 , β((r/K)+β)I* -(r/K)S*α1 [approximate]6.6343>0 , that is the condition (H2) holds. We can get (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]-p32 (p0 +q0 )[approximate]23.1510>0 ,h3 [approximate]6.0704>0 , h2 [approximate]6.5263>0, h1 [approximate]1.6535>0 , and p0 -q0 [approximate]0.1310>0 . By Theorem 4(i), the coexistence equilibrium E* is locally asymptotically stable, for all τ...5;0 . Numerical simulation illustrates our result (see Figure 1).
Figure 1: The temporal solution found by numerical integration of system (2) with τ=1 .
[figure omitted; refer to PDF]
Example 6.
In (2), we let a=0.55, p=0.95, r=20, r1 =0.5, k=1, β=1, d=d1 =d2 =0.1, b=1 , and m=0.5 . System (2), with the above coefficients, has a unique coexistence equilibrium E* (0.8946,0.1007,0.1266,0.6332) . It is easy to show that (r/K)S* -α1 [approximate]17.7848>0 , and β((r/K)+β)I* -(r/K)S*α1 [approximate]0.1083>0 ; that is, the condition (H2) holds. We can get (p1 +q1 )[p3 (p2 +q2 )-(p1 +q1 )]-p32 (p0 +q0 )[approximate]199.2429>0, h3 [approximate]316.4769>0, h2 [approximate]116.9725>0, h1 [approximate]1.1233>0 , and p0 -q0 [approximate]-0.0875<0 . By Theorem 4(ii), there exists a positive number τ0 [approximate]11.4092 , such that the coexistence equilibrium E* is locally asymptotically stable if 0...4;τ<τ0 and is unstable for τ>τ0 . Numerical simulations illustrate our results (see Figure 2).
The temporal solution found by numerical integration of system (2) with (a) τ=1 and (b) τ=15 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
Now, we are concerned with the global attractiveness of the coexistence equilibrium E* .
Theorem 7.
The coexistence equilibrium E* (S* ,I* ,Y1* ,Y2* ) of system (2) is globally attractive provided that the following conditions hold:
(i) 0<(1-m)<(a/b)(Kβ-d) ;
(ii) d2 (r1 +d1 )<r1 pb<2d2 (r1 +d1 ) .
That is, the system (2) is persistent, if conditions (i) and (ii) hold.
Proof.
Let (S(t),I(t),Y1 (t),Y2 (t)) be any positive solution of system (2) with initial conditions (3). Let [figure omitted; refer to PDF] We now claim that US =LS =S* , UI =LI =I* , UYi =LYi =Yi* (i=1,2) . The strategy of the proof is to use an iteration technique.
We derive from the first and the second equations of the system (2) that [figure omitted; refer to PDF] Consider the following auxiliary equations: [figure omitted; refer to PDF] If Kβ>d , then by Theorem 3.1 in [4], it follows from (47) that [figure omitted; refer to PDF] By comparison, we obtain that [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T1 >0 such that if t>T1 , then I(t)...4;M1I +[straight epsilon] .
It follows from the third and the fourth equations of system (2) that, for t>T1 +τ , [figure omitted; refer to PDF] Consider the following auxiliary equations: [figure omitted; refer to PDF] If r1 pb>d2 (r1 +d1 ) , then by Lemma 2.4 in [9], it follows from (51) that [figure omitted; refer to PDF] By comparison, we obtain that [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T2 >T1 +τ such that if t>T2 , then Y2 (t)...4;M1Y2 +[straight epsilon] .
We derive from the first and the second equations of system (2) that [figure omitted; refer to PDF] Since (1-m)<(a/b)(Kβ-d) holds, by Theorem 3.1 in [4], it follows from (54) and comparison argument that [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T3 >T2 such that if t>T3 , then I(t)...5;N1I -[straight epsilon] . We derive from the third and the fourth equations of system (2) that, for t>T3 +τ , [figure omitted; refer to PDF] Since r1 pb>d2 (r1 +d1 ) holds, by Lemma 2.4 of [9], it follows from (56) and comparison argument that [figure omitted; refer to PDF] Since these two inequalities hold, for arbitrary [straight epsilon]>0 sufficiently small, we conclude that LY1 ...5;N1Y1 , LY2 ...5;N1Y2 , where [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T4 ...5;T3 +τ , such that if t>T4 , Y2 (t)...5;N1Y2 -[straight epsilon] .
For [straight epsilon]>0 sufficiently small, we derive from the first and the second equations of system (2) that, for t>T4 , [figure omitted; refer to PDF] By comparison and Theorem 3.1 in [4], it follows that [figure omitted; refer to PDF] Since these two inequalities hold, for arbitrary [straight epsilon]>0 sufficiently small, we conclude that US ...4;M2S , UI ...4;M2I , where [figure omitted; refer to PDF] Therefore, for [straight epsilon]>0 sufficiently small, there is a T5 ...5;T4 such that if t>T5 , I(t)...4;M2I +[straight epsilon] .
For [straight epsilon]>0 sufficiently small, we derive from the third and the fourth equations of system (2) that, for t>T5 +τ , [figure omitted; refer to PDF] Since pbr1 >d2 (r1 +d1 ) holds, by Lemma 2.4 of [9], it follows from (62) that [figure omitted; refer to PDF] Since these two inequalities hold, for arbitrary [straight epsilon]>0 sufficiently small, we conclude that UY1 ...4;M2Y1 , UY2 ...4;M2Y2 , where [figure omitted; refer to PDF] Therefore, for [straight epsilon]>0 sufficiently small, there is a T6 ...5;T5 +τ such that if t>T6 , y2 (t)...4;M2Y2 +[straight epsilon] .
For [straight epsilon]>0 sufficiently small, it follows from the first and the second equations of system (2) that, for t>T6 , [figure omitted; refer to PDF] By Theorem 3.1 in [4] and comparison argument, we can obtain [figure omitted; refer to PDF] Since these two inequalities hold, for arbitrary [straight epsilon]>0 sufficiently small, we conclude that LS ...5;N2S , LI ...5;N2I , where [figure omitted; refer to PDF] Hence, for [straight epsilon]>0 sufficiently small, there is a T7 ...5;T6 such that if t>T7 , I(t)...5;N2I -[straight epsilon] . We therefore obtain from the third and the fourth equations of system (2) that, for t>T7 +τ , [figure omitted; refer to PDF] Since pbr1 >d2 (r1 +d1 ) holds, by Lemma 2.4 in [9] and comparison argument, we derive that [figure omitted; refer to PDF] Since these inequalities hold for arbitrary [straight epsilon]>0 sufficiently small, we conclude that LY1 ...5;N2Y1 , LY2 ...5;N2Y2 , where [figure omitted; refer to PDF] Continuing this process, we derive eight sequences MkS , MkI , MkY1 , MkY2 , NkS , NkI , NkY1 , and NkY2 (k=1,2,...) such that, for k...5;2 , [figure omitted; refer to PDF] It is readily seen that [figure omitted; refer to PDF] Noting that the sequences MkS , MkI , MkY1 , MkY2 are nonincreasing, and the sequences NkS , NkI , NkY1 , NkY2 are nondecreasing. Hence, the limit of each sequence in MkS , MkI , MkY1 , MkY2 , NkS , NkI , NkY1 , and NkY2 exists. Denote [figure omitted; refer to PDF] From (71), we can obtain [figure omitted; refer to PDF] It follows from (74) that [figure omitted; refer to PDF] (75) minus (76), [figure omitted; refer to PDF] Assume that I¯...0;I_ . Then we derive from (77) that [figure omitted; refer to PDF] (75) plus (76), [figure omitted; refer to PDF] On substituting (78) into (79), it follows that [figure omitted; refer to PDF] Note that I¯>0 and I_>0 . If d2 (r1 +d1 )<r1 pb<2d2 (r1 +d1 ) , we derive that 1-m>a(Kβ-d)/b . This is a contradiction. Hence, we have S¯=S_ . It therefore follows from (74) that I¯=I_, Y¯1 =Y_1 , and Y¯2 =Y_2 . We therefore conclude that E* is globally attractive. The proof is complete.
5. Conclusion
In this paper, we have incorporated a prey refuge, stage structure for the predator and time delay due to the gestation of the predator into a predator-prey system. Incorporating a refuge into system (1) provides a more realistic model. A refuge can be important for the biological control of a pest; however, increasing the amount of refuge can increase prey densities and lead to population outbreaks. By using the iteration technique and comparison arguments, respectively, we have established sufficient conditions for the global stability of the predator-extinction equilibrium and the globally attractivity for the coexistence equilibrium. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorem 3, we see that the predator population go to extinction if 0<(1-m)<(a/b)(Kβ-d) and kβ>d , d2 (r1 +d1 )>pbr1 . By Theorem 7, we see that if 0<(1-m)<(a/b)(Kβ-d) and d2 (r1 +d1 )<r1 pb<2d2 (r1 +d1 ) , then both the prey and predator species of system (2) are permanent.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (11101117).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] W. O. Kermack, A. G. Mckendrick, "A contribution on the mathematical theory of epidemics," Proceedings of the Royal Society A , vol. 115, pp. 700-721, 1927.
[2] R. M. Anderso, R. M. May Intections Disease of Humans Dynamics and Control , Oxford University Press, Oxford, UK, 1991.
[3] A. K. Pal, G. P. Samanta, "Stability analysis of an eco-epidemiological model incorporating a prey refuge," Nonlinear Analysis: Modelling and Control , vol. 15, no. 4, pp. 473-491, 2010.
[4] S. Wang The research of eco-epidemiological of models incorporating prey refuges [Ph.D. thesis] , Lanzhou University, 2012.
[5] Y. Xiao, L. Chen, "A ratio-dependent predator-prey model with disease in the prey," Applied Mathematics and Computation , vol. 131, no. 2-3, pp. 397-414, 2002.
[6] W. Wang, L. Chen, "A predator-prey system with stage-structure for predator," Computers & Mathematics with Applications , vol. 33, no. 8, pp. 83-91, 1997.
[7] Y. N. Xiao, L. S. Chen, "Global stability of a predator-prey system with stage structure for the predator," Acta Mathematica Sinica , vol. 20, no. 1, pp. 63-70, 2004.
[8] R. Xu, Z. Ma, "Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator," Nonlinear Analysis: Real World Applications , vol. 9, no. 4, pp. 1444-1460, 2008.
[9] R. Xu, Z. Ma, "Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure," Chaos, Solitons & Fractals , vol. 38, no. 3, pp. 669-684, 2008.
[10] W. G. Aiello, H. I. Freedman, "A time-delay model of single-species growth with stage structure," Mathematical Biosciences , vol. 101, no. 2, pp. 139-153, 1990.
[11] Y. Kuang, J. W.-H. So, "Analysis of a delayed two-stage population model with space-limited recruitment," SIAM Journal on Applied Mathematics , vol. 55, no. 6, pp. 1675-1696, 1995.
[12] J. Hale Theory of Functional Differential Equations , Springer, New York, NY, USA, 1977., 2nd.
[13] Y. Kuang Delay Differential Equations with Applications in Population Dynamics , vol. 191, Academic Press, New York, NY, USA, 1993.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Copyright © 2014 Lingshu Wang and Guanghui Feng. Lingshu Wang et al. 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.
Abstract
A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, when τ=[subscript]τ0[/subscript] . By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer