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An impulsive predator–prey model with communicable disease in the prey species only
Nonlinear Analysis: Real World Applications 10 (2009) 3098–3111
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
An impulsive predator–prey model with communicable disease in the prey species only Maoxing Liu a , Zhen Jin a , Mainul Haque b,∗ a
Department of Mathematics, North University of China, Taiyuan, Shanxi, 030051, PR China
b
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
article
info
Article history: Received 9 December 2007 Accepted 8 October 2008 Keywords: Eco-epidemiology Susceptible prey Infected prey Predator Impulsive effect Bifurcation Coincidence degree
a b s t r a c t A system of impulsive differential equations describing predator–prey dynamics with impulsive effect is proposed and analyzed with the assumption that a transmissible disease is spreading among the prey species only. At first, the ‘‘semi-trivial’’ periodic solution (S (t ), 0, 0) is given. After that, the existence of ‘‘infection-free’’ periodic solution (S (t ), 0, P (t )) and the ‘‘predator-free’’ periodic solution have been obtained via bifurcation. Finally, the method of coincidence degree has been used to derive a set of sufficient conditions for the existence of at least one strictly positive periodic solution. Numerical simulations and a brief discussion conclude the paper. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction After the pioneering work of Vito Volterra and Alfred James Lotka in the middle of twentieth century for predator–prey interactions, the seminal models and competitive system have been studied extensively during the last couple of decades. On the other hand, after the fundamental work of Kermack–McKendrick on SIRS (Susceptible–Infected–Recover–Susceptible) epidemiological models have also received much attention from a group of scientists. But in recent times, these two major fields of study are merged and renamed as a new field of study called ‘‘eco-epidemiology’’. A good number of studies have already been performed in this particular direction (see Hadeler and Freedman [1], Freedman [2], Beltrami and Carroll [3], Xiao and Chen [6,7], Hethcote et al. [8], Haque and his colleagues [4,5,23–27] and the references therein). Impulsive methods have been applied in almost every field of applied sciences. The theoretical investigations and their applications can be found in [9,10]. Recently, the impulsive effect on the ecological situations have been studied by several researchers, for instance, a classical periodic Lotka–Volterra predator–prey system with impulsive effect is investigated by Tang and Chen [11]. Bing Liu and her co-workers have shown the dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management [12]. Similarly, at the same time, the role of impulsive effect on epidemic models has also been investigated; for example, Shulgin et al. have analyzed an SIR epidemic model with pulse vaccination strategy [13]. Alberto d’Onofrio has studied the stability properties of pulse vaccination strategy in SIR and SEIR epidemic models [14,15]. But we know so far there has been no work that dealt with the impulsive effect on the eco-epidemiological system. This is a first attempt to look for the influence of impulsive effect on an eco-epidemiological system. The first eco-epidemiological model has been considered by Hadeler and Freedman [1] in which a disease was spreading among interacting populations. In Venturino [26] the Lotka–Volterra model is taken as the demographic basis to study the influence of a disease propagating in one of the two species. In an aquatic medium, a model of this kind has been first
∗
Corresponding author. Tel.: +44 0 115 951 4949; fax: +44 0 115 951 4951. E-mail address: [email protected] (M. Haque).
1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.010
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introduced in Beltrami and Carroll [3]. Jin and Haque [25] have shown the global stability analysis of an eco-epidemiological model of the Salton sea, the largest lake of California. This lake has been facing problems due to its massive fish and bird mortality. Four types of fish are very common in the Salton Sea, namely Tilapia (Oreochromis mossambicus), Corvina (Cynoscian Xanthulus), Coraker (bairdiella icistius) and Sargo (Anisopremus davidsoni). The actual cause of death is still a mystery, although a toxic algal bloom is being investigated as a possible answer. A transmissible disease is running through the prey species (fish). The fish-eating birds feed on them. At the end of the article, the authors are able to show that for global persistence of the prey and predator (fish-eating birds like ‘‘Pelicans’’) populations from the explosive epidemic in the Salton Sea, reduction of the prey population (especially the infected fishes) in a considerable amount is essential, and hence suitable harvesting policy must be implemented. Their findings are also supported by experimental results [28]. Now in order to apply impulsive effect on this kind of situation one could ask a question; what will happen if people catch these fish impulsively according to seasons? In order to get a proper understanding of this kind of question, we have proposed and analyzed an eco-epidemiological model with impulsive effect. Nevertheless, with the pulse vaccination strategy like [13], one could incorporate the impulsive effect on eco-epidemic situations. Therefore our attempt will open a new window of research in this direction. The article is organized as follows: we have proposed an eco-epidemiological model with impulsive effect in Section 2; the existence and the stability of the ‘‘semi-trivial’’ solution is given in Section 3. In Section 4, we have analyzed the existence and the bifurcation of the ‘‘infection-free’’ solution and the ‘‘predator-free’’ solution, and the existence of a positive periodic solution is analyzed in Section 5. 2. An eco-epidemiological model with impulsive effect 2.1. The basic ecological assumptions (A1) In the absence of transmissible disease the prey population grows according to the logistics law with carrying capacity K (K ∈ R+ ) and intrinsic birth rate constant γ (γ ∈ R+ ). (A2) In the presence of disease, the prey population is divided into two classes, namely susceptible prey, denoted by S (t ), and infected prey, denoted by I (t ). (A3) The result of Bertheir et al. [16] shows that a mass-action incidence assumption is more appropriate than a proportional mixing one in describing the dynamics of direct transmission. Again data of Greenwood [17] experiment suggests that there is no change of qualitative properties upon the contact process whether it follows the law of mass action or proportional mixing. Therefore we assume that the transmission of the disease follow the simple law of mass action λSI like [23,24], where λ is the force of infection. (A4) We assume that only susceptible prey is capable of reproducing and contributing to its carrying capacity. We also assume that the infected prey do not grow and reproduce, but the infected prey can be recovered. The model of Hamilton et al. [18] assumes that infected individuals do not reproduce, infection rather reduces the remaining capacity due to inability to compete for resources. (A5) Nowadays it has been observed that viral, bacterial, fungal parasites can mediate hosts vulnerable to predation [19]. The review of Holmes and Bethel [20] contains many examples in which the parasites change the external features or behavior of the prey, so infected prey may live in the locations that are more accessible to predator. Also, due to infection, they become less active and therefore could be caught easily by the predator compared to susceptible individuals. Thus we assume that searching coefficient of the predator for infected prey is greater than that of susceptible prey. 2.2. The basic differential equations With the above assumptions we get the following set of autonomous non-linear differential equations:
S 0 S = γ S 1 − − c1 SP − λSI + µI , t 6= tn . K 0 I = λSI − c2 IP − µI − δ1 I , tn+1 = tn + T , P 0 = e(c1 SP + c2 IP ) − δ2 P . S (t + ) = (1 − h1 )S (t − ), t = tn . + − I (t +) = (1 −− h2 )I (t ), n = 0, 1, 2 . . . , P (t ) = P (t ).
(2.1)
Here, system (2.1) has to be analyzed with the following initial conditions: S (0) = S0 > 0;
I (0) = I0 > 0;
P (0) = P0 > 0;
(2.2)
where µ is the recover rate, δ1 , the total death rate of the infected prey (the natural death rate of the prey and the death rate of the prey due to infection), δ2 , the death rate of predator population, e, the conversion factor, and c1 and c2 are the searching coefficients of the predator for susceptible and infected prey respectively. We assume that c2 ≥ c1 .
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Let us assume that in a single pulse, the pulse scheme proposes to control a fraction h1 of susceptible prey and a fraction h2 of the infected prey, which is being applied for every T time. We also assume that due to this pulse, the amount of prey leaving from the poll of susceptible and infected individuals are respectively h1 S and h2 I. 3. Existence of the ‘semi-trivial’ periodic solution Let us consider the following single Logistic model with impulsive effect:
S
, t 6= tn , tn+1 = tn + T , K S (t ) = (1 − h1 )S (t − ), t = tn , n = 0, 1, 2 . . . . S0 = γ S
1−
(3.1)
+
It is easy to show that system (3.1) admits a positive periodic solution if and only if T > τ0 = unique and stable if it exists. In fact, for system (3.1), we carry out the change of variable x = linear non-homogeneous impulsive equation
(
− ln(1−h1 ) . γ
Moreover, it is
and obtain the following
γ
, t 6= tn , tn+1 = tn + T , K x(t ) = (1 − h1 )x(t − ), t = tn , n = 0, 1, 2 . . . . x0 = −γ x +
1 , S
+
(3.2)
Let
Y
W (t , s) =
s≤τk
1 1 − h1
exp(−γ (t − s))
(3.3)
be the Cauchy matrix for the respective homogeneous equation. Then x(t ) = W (t , 0)x(0) +
γ
t
Z
K
W (t , s)ds
(3.4)
0
is a solution of system (3.2). This solution is T -periodic if x(0) = x(T ), or if
(1 − W (T , 0))x(0) =
γ K
T
Z
W (t , s)ds.
(3.5)
0
Since the multiplier W (T , 0) of the homogeneous equation x0 = −γ x, t 6= tn , tn+1 = tn + T , x(t + ) = (1 − h1 )x(t − ), t = tn , n = 0, 1, 2 . . . ,
− ln(1−h )
γ
(3.6)
RT
1 is less than 1 because we have γ > , and K 0 W (t , s)ds > 0. Eq. (3.6) has a unique solution x(0) = p0 > 0. T By using the initial condition x(0) = p0 , one could obtain the corresponding unique positive T -periodic solution of system (3.2), and then the unique T -periodic solution of system (3.1) could be derived from there. Obviously this solution is also positive and stable. Thus we have done.
4. Existence of the ‘infection-free’ periodic solution and the ‘predator-free’ solution 4.1. Preliminaries In this section, we followed the technique used by A. Lakmeche and O. Arino in [21], here we gave the existence of the ‘infection-free’ and the ‘predator-free’ periodic solutions. We introduced some notations and stated some preliminary results. Let us consider the following impulsive system given below
0 S (t ) = F1 (S (t ), P (t )), t 6= tn , 0 P (t ) = F2 (S (t ), P (t )), tn+1 = tn + T , + S (t +) = θ1 (S (t ), P (t )), t = tn , P (t ) = θ2 (S (t ), P (t )), n = 0, 1, 2 . . . .
(4.1)
Suppose that
S 0 (t ) = F1 (S (t ), 0), t 6= tn , tn+1 = tn + T , S (t + ) = θ1 (S (t ), 0), t = tn , n = 0, 1, 2 . . .
(4.2)
which is obtained by taking P (t ) = 0 in (4.1), has a periodic solution denoted by U˜ (t ). Denote a trivial periodic solution of system (4.1) by π = (U˜ , 0)T . Letting Φ be the flow associated to system (4.1), we have U (t ) = Φ (t , S0 , P0 ), 0 < t ≤ T ,
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where U0 = U (S0 , P0 ). The flow Φ applies to time T . So, U (T ) = Φ (T , U0 ). The following notations of [21] will be used:
∂θ2 ∂ Φ2 d0 = 1 − (τ0 , U0 ), ∂P ∂P ∂θ1 ∂ Φ1 a00 = 1 − (τ0 , U0 ), ∂S ∂S ∂θ1 ∂ Φ1 b00 = − (τ0 , U0 ), ∂S ∂P ∂ 2 θ2 ∂ Φ1 (τ0 , U0 ) ∂ Φ1 (τ0 , U0 ) 1 ∂θ1 ∂ Φ1 (τ0 , U0 ) ∂ Φ2 (τ0 , U0 ) B = − + ∂ S∂ P ∂τ ∂S a00 ∂ S ∂τ ∂P 2 ∂θ2 ∂ Φ2 (τ0 , U0 ) 1 ∂θ1 ∂ Φ1 (τ0 , U0 ) , − ∂P ∂ S∂ P a00 ∂ S ∂τ 0 ∂ 2 θ2 b0 ∂ Φ1 (τ0 , U0 ) ∂ Φ1 (τ0 , U0 ) ∂ Φ2 (τ0 , U0 ) ∂ 2 θ2 ∂ Φ2 (τ0 , U0 ) 2 C = −2 − 0 − ∂ S∂ P a0 ∂S ∂P ∂P ∂P2 ∂P 0 2 2 ∂θ2 b0 ∂ Φ2 (τ0 , U0 ) ∂θ2 ∂ Φ2 (τ0 , U0 ) − , +2 ∂ P a00 ∂ S∂ P ∂I ∂P2 0
where τ0 is the root of d00 = 0, and
Z t ∂ Φ1 (t , U0 ) ∂ F1 (π (ξ )) = exp dξ , ∂S ∂S 0 Z t ∂ F2 (π (ξ )) ∂ Φ2 (t , U0 ) = exp dξ , ∂P ∂P 0 Z t Z v Z t ∂ Φ1 (t , U0 ) ∂ F1 (π (ξ )) ∂ F1 (π (v)) ∂ F2 (π (ξ )) = exp dξ exp dξ dv, ∂P ∂S ∂P ∂P 0 v 0 Z t 2 Z v Z t 2 ∂ Φ2 (t , U0 ) ∂ F2 (π (ξ )) ∂ F2 (π (v)) ∂ F2 (π (ξ )) = exp dξ exp dξ dv, ∂P∂S ∂P ∂P∂S ∂P 0 v 0 Z t 2 Z v Z t 2 ∂ F2 (π (ξ )) ∂ F2 (π (ξ )) ∂ Φ2 (t , U0 ) ∂ F2 (π (v)) = exp d ξ exp d ξ dv, 2 ∂P2 ∂P ∂P 0 ∂ P2 0 Z t v Z t ∂ F2 (π (ξ )) ∂ F2 (π (v)) + exp dξ ∂ P 0 Z Zv v ∂P∂S Z θ v ∂ F1 (π (ξ )) ∂ F1 (π (v)) ∂ F2 (π (ξ )) × exp dξ exp dξ dθ dv, ∂P ∂P ∂P 0 θ 0 Z t ∂ 2 Φ2 (t , U0 ) ∂ F2 (π (v)) ∂ F2 (π (ξ )) = exp dξ , ∂ P ∂τ ∂P ∂P 0 ∂ Φ1 (τ0 , U0 ) = S˜ 0 (τ0 ), ∂τ then we get the following lemma. Lemma 4.1. If |1 − a00 | < 1 and d00 = 0, then we get: (a) If BC 6= 0, then we have a bifurcation. Moreover, we have a bifurcation of a nontrivial periodic solution of system (4.1) if BC < 0, and a subcritical case if BC > 0. (b) If BC = 0, then we have an undetermined case. 4.2. Existence of the ‘infection-free’ periodic solution Let us consider the following set of differential equations
S S0 = γ S 1 − − c1 SP , F1 (S , P ), K 0 P = ec1 SP − δ2 P , F2 (S , P ), + S h1 )S (t − ) , θ1 (S , P ), (t +) = (1 − − P (t ) = P (t ) , θ2 (S , P ),
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . .
(4.3)
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Fig. 1. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, δ1 = 0.02, δ2 = 0.0139, h1 = 0.3, T = 2.
which satisfy F2 (S , 0) = θ2 (S , 0) ≡ 0, θ1 6= 0 for S 6= 0 and θ2 6= 0 for P 6= 0. In order to apply Lemma 4.1, we compute the following: d00 = 1 − e
R τ0 ∂ F2 (λ(ξ )) ∂I
0
If d0 = 0, we can get τ0 = In addition, 0
dξ
= 1 − e−δ2 τ0 +λ
b00 = −(1 − h1 )
0
s˜(ξ )dξ
.
−ec1 K ln(1−h1 ) . γ (ec1 K −δ2 )
a00 = 1 − (1 − h1 )eγ τ0 − τ0
Z
R τ0
2γ K
0
s˜(ξ )dξ
> 0,
2γ 0 ν γ − K s˜(ξ ) dξ
Rτ
e
R τ0
(−c1 s˜(ν))e(−δ2 τ0 )+ec1
R τ0 0
s˜(ξ )dξ
dν > 0.
0
∂θ
∂θ
∂θ
∂θ
∂2θ
∂2θ
Since ∂ P1 = ∂ S2 = 0, ∂ S1 = 1 − h1 , ∂ P2 = 1, ∂ P 22 = ∂ S ∂ 2P = 0, it is easy to verify that C > 0, and if ec1 then R τ0 (1 − h1 )(1 − h2 )S˜ 0 (τ0 ) ∂ 2 Φ2 (τ0 , U0 ) B=− + [ec1 S˜ (τ0 ) − δ2 ]e 0 · 0 a0 ∂ S∂ I
∂ F2 (λ(ξ )) dξ ∂P
R τ0 0
S˜ (ξ )dξ − δ2 τ0 > 0,
! < 0.
According to Lemma 4.1, we get the following result. Theorem 4.1. System (2.1) has a positive periodic ‘infection-free’ solution if T > τ1 and is close to τ1 , where τ1 satisfies
τ1 =
−ec1 K ln(1 − h1 ) . γ (ec1 K − δ2 )
(4.4)
This theorem illustrates that the ‘semi-trivial’ solution becomes unstable if the period T is more than τ1 and close to τ1 , and become the ‘infection-free’ solution. Numerical simulations confirm our analytical results (Figs. 1 and 2). 4.3. Existence of the ‘predator-free’ periodic solution Let us consider the following set of differential equations
γ 2 ˙ S = γ S − K S − λSI + µI , F1 (S , I ), ˙I = −δ1 I + λSI − µI , F2 (S , I ), S (t + ) = (1 − h1 )S (t − ) , θ1 (S , I ), + I (t ) = (1 − h2 )I (t − ) , θ2 (S , I ),
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . .
with F2 (S , 0) = θ2 (S , 0) ≡ 0, θ1 6= 0 for S 6= 0 and θ2 6= 0 for I 6= 0. We compute the following: d00 = 1 − e
R τ0 ∂ F2 (λ(ξ )) 0
∂I
If d0 = 0, we can get τ0 = 0
dξ
= 1 − e(−(δ1 +µ)τ0 )+λ
−λK ln(1−h1 ) . γ (λK −δ1 −µ)
R τ0 0
s˜(ξ )dξ
.
(4.5)
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Fig. 2. Time-series for the P (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, δ2 = 0.0139, h1 = 0.3, T = 2.
In addition, a00 = 1 − (1 − h1 )eγ τ0 − b00 = −(1 − h1 )
τ0
Z
e
2γ K
R τ0 0
s˜(ξ )dξ
> 0,
Rτ 2γ 0 ν γ − K s˜(ξ ) dξ
(−λ˜s(ν) + µ)e(−(δ1 +µ)τ0 )+λ
R τ0 0
s˜(ξ )dξ
dν > 0 .
0
∂θ
∂θ
∂θ
∂θ
∂2θ
∂2θ
Since ∂ I1 = ∂ S2 = 0, ∂ S1 = 1−h1 , ∂ I2 = 1−h2 , ∂ I 22 = ∂ SI2 = 0, it is easy to verify that C > 0, and if λ then R τ0 (1 − h1 )(1 − h2 )S˜ 0 (τ0 ) ∂ 2 Φ2 (τ0 , U0 ) · + [λS˜ (τ0 ) − δ1 ]e 0 B=− 0 a0 ∂ S∂ I
∂ F2 (λ(ξ )) dξ ∂I
R τ0 0
S˜ (ξ )dξ −δ1 τ0 < 0,
! < 0.
According to Lemma 4.1, we get the following result. Theorem 4.2. System (2.1) has a positive periodic ‘predator-free’ solution if T > τ2 and is close to τ2 , where τ2 satisfies
τ2 =
−λK ln(1 − h1 ) . γ (λK − δ1 − µ)
(4.6)
This theorem illustrates that the ‘semi-trivial’ solution becomes unstable if the period T is more than τ2 and close to τ2 , the variable I (t ) begins to oscillate with an amplitude, and become the ‘predator-free’ solution. Numerical simulations confirm our theoretical results (Figs. 3 and 4). 5. Existence of the positive T -periodic solution We assume µ = 0 through out this section. Before using the method of coincidence degree, we shall mention some definitions and basic concepts from the book of Gaines and Mawhin [22], interested readers are requested to refer the book. Let X , Z be real Banach spaces, L : Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z . If L is a Fredholm mapping of index zero there exist continuous projects P : X → X and Q : Z → Z such that Im P = Ker L, Ker Q = Im L = Im(I − Q ). It follows that L |Dom L T Ker P : (I − P )X → Im L is invertible. We denote the inverse of that
¯ if QN (Ω ¯ ) is bounded and mapping by Kp . If Ω is an open bounded subset of X , the mapping N will be called L-compact on Ω ¯ Kp (I − Q )N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exit isomorphisms J : Im Q → Ker L. In the proof of the existence theorem below, we will use the continuation theorem from Gaines and Mawhin [22]. ¯ . Suppose Lemma 5.1. Let L be a Fredholm mapping of index zero and let N be L-compact on Ω ¯ Ω. (a) For each β ∈ (0, 1), every solution x of Lx = β Nx is such that x∈∂ (b) QNx = 6 0 for each x ∈ ∂ Ω ∩ Ker L and deg{JQN , Ω ∩ Ker L, 0} 6= 0. ¯. Then the equation Lx = Nx has at least one solution which belongs to Dom L ∩ Ω
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Fig. 3. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, λ = 0.4, u = 0.01, δ1 = 0.02, h1 = 0.3, h2 = 0.1, T = 1.
Fig. 4. Time-series for the I (t ) evolving according to the model with γ = 0.5, K = 10, λ = 0.4, u = 0.01, δ1 = 0.02, h1 = 0.3, h2 = 0.1, T = 1.
Theorem 5.1. If γ T + ln(1 − h1 ) > 0 and
λδ2 K > 0, γ T γ c2 T γ ec2 ln(1 − h2 ) δ2 c1 K ln(1 − h2 ) Kc 2 λδ2 c1 K − γ+ + 12 δ2 − + > 0, ec2 c2 γ T T γ c2 γ ec22 ln(1 − h2 ) λK ln(1 − h1 ) λKc1 ln(1 − h2 ) λ2 δ2 K −δ1 + + γ+ + δ − − > 0, 2 2 T c2 γ T T γ c2 γ ec22 K
γ+
ln(1 − h1 )
+
Kc1
δ2 −
ln(1 − h1 )
−
then system (2.1) has at least one positive T -periodic solution. Proof. Making the change of variable S (t ) = es(t ) ,
I (t ) = ei(t ) ,
P (t ) = ep(t )
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then system (2.1) is transformed into,
γ s˙(t ) = γ − es(t ) − λei(t ) − c1 ep(t ) , t 6= tn . K s(t ) p(t ) ˙i(t ) = −δ1 + λe − c2 e , tn+1 = tn + T , s(t ) p˙ (t ) = −δ2 + e(c1 e + ec2 ei(t ) ). + − s ( t ) − s ( t ) = ln ( 1 − h1 ), t = tn . + − i ( t ) − i ( t ) = ln ( 1 − h n = 0, 1, 2 . . . , 2 ), + p(t ) − p(t − ) = 0.
(5.1)
Let Dom L = PCT1 × PCT1 and s L : Dom L → Z , i p
! ∆s(tk ) ∆i(tk ) . ∆p(tk )
s˙ → ˙i , p˙
!
(5.2)
Again assume Dom N = PCT1 × PCT1 → Z with
N
s i p
!
γ−
γ
es(t ) − λei(t ) − c1 ep(t )
!
K −δ1 + λes(t ) − c2 ep(t ) , −δ2 + e(c1 es(t ) ) + ec2 ei(t )
→
ln(1 − h1 ) ln(1 − h2 ) . 0
(5.3)
Obviously,
( Ker L =
s i p
s i p
! :
! =
C1 C2 C3
!
) ∈ R , t ∈ [0, T ] 3
(5.4)
and
Im L =
Z
T
0 ! ! ! l ak Z T m , bk ∈Z : m + bk T = 0 . Z0 n ck T n + ck T = 0 l + ak T = 0
z=
(5.5)
0
Since Im L is closed in Z , L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ),
where
Z s i p
!
P
T
l + ak T = 0
0 Z T = m + bk T = 0 , T 0 Z T n + ck T = 0 1
(5.6)
0
and
Z l ak m , bk n ck
!
QZ = Q
!!
1 = T
T
l + ak T = 0
0 Z T 0! m + bk T = 0 , 0 . 0 0 Z T n + ck T = 0 0
(5.7)
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Furthermore, the generalized inverse Kp : Im L → Ker P ∩ Dom L is given by
Z
t
l + ak − Z t0 Kp Z = m + bk − Z0 t n + ck − 0
1
T
Z
t
Z
T Z0 Z0 T t 1
l − ak
m − bk . T Z0 Z0 T t 1 n − ck T
0
(5.8)
0
Thus
Z T Z Z γ s(t ) 1 T i(t ) 1 T 1 1 p(t ) λ e − c e + e − ln ( 1 − h ) γ − 1 1 ! T 0 K T 0! Z T T 0 Z T T 0 s 1 1 1 , 0 , QN i = λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ1 + T 0 0 p Z T T 0 Z TT 1 1 −δ2 + e(c1 es(t ) ) + ec2 ei(t ) T
T
0
(5.9)
0
and
s i p
!
Kp (I − Q )N
=
−
γ
Z t λei(t ) − c1 ep(t ) + ln(1 − h1 ) 0 0 K Z Z t 0 t −δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) 0 0 Z t Z t s(t ) i(t ) −δ2 + e(c1 e ) + ec2 e 0 0 1 Z t h i γ s(t ) γ − e − λei(t ) − c1 ep(t ) + ln(1 − h1 ) T 0 Z K t 1 [−δ1 + λes(t ) − c2 ep(t ) ] + ln(1 − h2 ) T 0 Z t 1 s(t ) i(t ) [−δ2 + ec1 e + ec2 e ]
γ−
Z
t
es(t ) −
T
Z
t
0
Z t h i 1 γ − γ − es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 ) T 2 K 0 Z t 1 1 − [−δ1 + λes(t ) − c2 ep(t ) ] + ln(1 − h2 ) T 2 0 Z t 1 1 − [−δ2 + ec1 es(t ) + ec2 ei(t ) ]
1 −
T
2
.
(5.10)
0
¯) Clearly, QN and Kp (I − Q )N are continuous. Using the Arzela–Ascoli Theorem, it is not difficult to show that Kp (I − Q )N (Ω is compact for any open bounded set Ω ⊂ X . The isomorphism J of Im Q onto Ker L may be defined by J : Im Q → X ,
h1 h2
,
0 0
→
h1 h2
.
(5.11)
Now corresponding to the operator equation s L i p
!
s i , p
!
= βN
(5.12)
we have
h i γ s˙(t ) = β γ − es(t ) − λei(t ) − c1 ep(t ) , K s(t ) p(t ) ˙i(t ) = β[−δ1 + λe − c2 e ], s(t ) p˙ (t ) = β[−δ2 + e(c1 e ) + ec2 ei(t ) ], + − s(t+ ) − s(t− ) = β ln(1 − h1 ), i(t +) − i(t −) = β ln(1 − h2 ), p(t ) − p(t ) = 0.
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . . ,
(5.13)
M. Liu et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3098–3111
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We prove the existence of M0 such that every T -periodic solution of system (2.1) satisfies |s(t )| + |i(t )| + |p(t )| ≤ M0 , let (s(t ), i(t ), p(t )) be a T -periodic solution. By integrating (5.13) we obtain
Z T h i γ s(t ) e + λei(t ) + c1 ep(t ) dt = γ T + ln(1 − h1 ), K Z0 T [λes(t ) − c2 ep(t ) ]dt = δ1 T − ln(1 − h2 ), 0 Z T [ec1 es(t ) + ec2 ei(t ) ]dt = δ2 T .
(5.14)
0
Since s(t ), i(t ), p(t ) ∈ X , there exist ξ¯ : s(ξ¯ ) = maxt ∈[0,T ] s(t ), η¯ : i(η) ¯ = maxt ∈[0,T ] i(t ), ζ¯ : p(ζ¯ ) = maxt ∈[0,T ] p(t ). On the other hand, there also exist ξ , η, ζ , such that: s(ξ + ) = inft ∈[0,T ] s(t ), i(η+ ) = inft ∈[0,T ] i(t ), p(ζ + ) = inft ∈[0,T ] p(t ). Here, if ξ , η, ζ is not an impulsive point, then we have s(ξ + ) = s(ξ ), i(η+ ) = i(η), p(ζ + ) = p(ζ ); if it is an impulsive point tk , then we have s(ξ + ) = s(tk+ ), i(η+ ) = i(tk+ ), p(ζ + ) = p(tk+ ). It follows from system (5.14) that
Z T |˙s(t )| ≤ 2(γ T + ln(1 − h1 )), Z0 T ¯ ¯ |˙i(t )| ≤ δ1 T + λT es(ξ ) + c2 eT es(ζ ) + ln(1 − h2 ), Z0 T |˙p(t )| ≤ 2δ2 T .
(5.15)
0
From (5.13) it follows that:
γ K
es(ξ ) + λei(η) + c1 e
p(ζ )
ln(1 − h1 )
≤γ +
T
,
(5.16)
then we can get
K ln(1 − h1 ) , s(ξ ) ≤ ln γ γ + T 1 ln(1 − h1 ) i(η) ≤ ln γ+ , λ T p(ζ ) ≤ ln 1 γ + ln(1 − h1 ) . c1
(5.17)
T
So s(t ) ≤ s(ξ ) +
T
Z
|˙s(t )| ≤ ln
0
K
γ
γ+
ln(1 − h1 )
T
+ 2(γ T + ln(1 − h1 )) , sM ,
ln(1 − h1 ) 1 ¯ ¯ |˙i(t )| ≤ ln γ+ + δ1 T + λT es(ξ ) + c2 eT es(ζ ) + ln(1 − h2 ) , iM , λ T 0 Z T 1 ln(1 − h1 ) p(t ) ≤ p(ζ ) + |˙p(t )| ≤ ln γ+ + 2δ2 T , pM . i(t ) ≤ i(η) +
Z
T
c1
0
T
We shall give here lower bound for sL , iL , pL as follows. From (5.14) we get
Z T λδ2 T γ c1 es(t ) dt = γ T + ln(1 − h1 ) + (δ2 T − ln(1 − h2 )) − , K c ec2 2 0 Z T Z T s(t ) λ e dt − c ep(t ) dt = δ1 T − ln(1 − h2 ), 2 0Z 0Z T T ec1 es(t ) dt + ec2 ei(t ) dt = δ2 T . 0
(5.18)
0
In fact, we obtain from the first equation of (5.18) s(ξ¯ ) ≥ ln
K
γ
γ+
ln(1 − h1 ) T
+
Kc1
γ c2
δ2 −
ln(1 − h1 ) T
−
λδ2 K , γ ec2
(5.19)
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then we get s(t ) ≥ s(ξ¯ ) −
T
Z
|˙s(t )|dt ≥ ln
K
γ+
ln(1 − h1 )
+
ln(1 − h1 ) λδ2 K δ2 − − γ c2 T γ ec2 Kc1
γ T 0 − 2(γ T + ln(1 − h1 )) , sL , Z T δ2 c1 K ln(1 − h2 ) Kc12 ln(1 − h2 ) λδ2 c1 K ˙ |i(t )|dt ≥ ln i(t ) ≥ i(η) ¯ − − (γ + ) + 2 δ2 − + ec2 c2 γ T T γ c2 γ ec22 0 s(ξ¯ ) s(ζ¯ ) − (δ1 T + λT e + c2 eT e + ln(1 − h2 )) , iL , Z T ln(1 − h2 ) λK ln(1 − h1 ) |˙p(t )|dt ≥ ln −δ1 + p(t ) ≥ p(ζ¯ ) − + γ+ T c2 γ T 0 λ2 δ2 K λKc1 ln(1 − h2 ) − + δ2 − − 2δ2 T , pL , T γ c22 γ ec22 therefore we obtain max |s(t )| ≤ max{|sM |, |sL |} , Ms ,
t ∈[0,T ]
max |i(t )| ≤ max{|iM |, |iL |} , Mi ,
t ∈[0,T ]
max |p(t )| ≤ max{|pM |, |pL |} , Mp .
t ∈[0,T ]
Clearly, Ms , Mi , Mp are independent of β , and M0 = Ms + Mi + Mp . By Lemma 5.1 and the assumptions in Theorem 5.1, it is not difficult to show that the system of algebraic equations
γ u + λv + c1 w = γ T + ln(1 − h1 ), K λu − c2 w = δ1 T − ln(1 − h2 ), ec u + ec v = δ T 1 2 2
(5.20)
has a unique solution (u∗ , v ∗ , w ∗ ) ∈ R3 . Let B be a ball of R3 centered at the origin of radius greater than γ T + ln(1 − h1 ), and
Ω = {X = (s, i, p) ∈ PCT1 × PCT1 × PCT1 : kxk < B}. It is clear that Ω verifies the requirement (a) in Lemma 5.1, when X ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R3 , X is a constant vector in R3 with kX k = B. Then
γ−
γ K
es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 )
,
−δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ2 + ec1 es(t ) + ec2 ei(t )
QNX =
0 0 0
!
and for X ∈ Ker L ∩ Ω we have
JQNX =
γ−
γ K
es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 )
−δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ2 + ec1 es(t ) + ec2 ei(t )
.
Furthermore, in view of assumption in Theorem 5.1, it is to easy to prove that
γ
deg {JQNX , Ω ∩ Ker L, 0} = sign det K λ ec1
λ 0 ec2
c1
∗ ∗ ∗ −c2 u v w 6= 0. 0
Therefore we have already proved that Ω satisfied all the requirements of Theorem 5.1. Hence, system (2.1) has at least ¯ , which implies that the result of Theorem 5.1 is true and thus the proof is complete. To one positive periodic solution in Ω substantiate our analytical finding we have performed the numerical simulations (Figs. 5–7).
M. Liu et al. / Nonlinear Analysis: Real World Applications 10 (2009) 3098–3111
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Fig. 5. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
Fig. 6. Time-series for the I (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
6. Discussion In this paper, we have proposed and analyzed a Lotka–Volterra type predator–prey system with disease in prey species with impulsive effect. A very similar model with out impulsive effect has been considered by Beltrami and Carroll [3] to present a phytoplankton–zooplankton system. They have compared their results with experimental results. Therefore we claim that our model is very much practical and widely used (as it is Lotka–Volterra type) in the field of ‘‘eco-epidemiology’’. At first, the ‘semi-trivial’ periodic solution (S (t ), 0, 0) has been shown, and then the existence of ‘infection-free’ periodic solution (S (t ), 0, P (t )) and ‘predator-free’ periodic solution near the ‘semi-trivial’ periodic solution are analyzed via bifurcation. Finally, by using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive periodic solution. From our analysis it can be observed that if the time period T is more than τ1 and close to τ1 , then impulsive ecoepidemiological model has the ‘infection-free’ solution. This is an important result from ecological view point since in this case only sound prey and the predator species co-exist and the transmissible disease dies out from the eco-system. So we have found out the range of time period under which the eco-system could be made disease-free. On the other hand, if the time period T is more than τ2 and close to τ2 , then susceptible prey and the infective prey are stable co-existent, that is under this conditions predator populations extinct. Therefore, considering both the situations, we can conclude that our model is able to search the value of time period T for which the predator population extinct and when it persist or in other
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Fig. 7. Time-series of the P (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
words, when the system will have infection and when it will be infection free. We would also like to mention that we have derived the conditions for which the solution of system (4.1) goes to a Hopf-bifurcation situation and also found out the conditions for which it is subcritical (Theorem 4.1). Finally, a set of sufficient conditions are obtained for the existence of at least one strictly positive periodic solution by using the method of coincidence degree. Note that in this case the three species, namely, sound prey, infected prey and the predator population exist in the eco-system. Our analytical results are confirmed by our numerical simulations. Before ending this article we would like to mention that this work is the first attempt to show the effect of impulse on eco-epidemiological model. So there is a room of improvement. We mention some future directions of research based on this article: (i) One may consider the situation where the predator is infected by eating infected prey. There should either be distinction made for healthy and infected predator, to account for the difference in mortality, or having the predator mortality dependent on the time-varying density of the infected prey. (ii) The time delay or latent period between contact, between susceptible and infected prey and the susceptible prey becoming infected could also be modelled and (iii) the infection rate may also be taken as ratio-dependent. These are the possible directions for future research. Acknowledgments The author Mainul Haque would like to thank to his teacher Professor John R. King, associate editor of SIAM J. Appl. Math., for his continuous encouragement and suggestions. A crucial part of this work was done when the third author Mainul Haque visited the North University of China, PR China, with the financial support of National Natural Science Foundation of China (10471040). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
K.P. Hadeler, H.I. Freedman, Predator-prey population with parasite infection, J. Math. Biol. 27 (1989) 609–631. H.I. Freedman, A model of predator–prey dynamics as modified by the action of a parasite, Math. Biosci. 99 (1990) 143–155. E. Beltrami, T.O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol. 32 (1994) 857–863. M. Haque, E. Venturino, An ecoepidemiological model with disease in predator: The ratio-dependent case, Math. Methods Appl. Sci. 30 (14) (2006) 1791–1809. E. Venturino, Epidemics in predator–prey model: Disease in the predators, IMA J. Math. Appl. Med. Biol. 19 (2002) 185–205. Y. Xiao, L. Chen, Modelling and analysis of a predator–prey model with disease in the prey, Math. Biosci. 171 (2001) 59–82. Y. Xiao, L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl. 258 (2) (2001) 733–754. H.W. Hethcote, W. Wang, L. Han, Z. Ma, A predator–prey model with infected prey, Theo. Pop. Biol. 66 (2004) 259–268. X.Z. Liu, Practical stabilization of control systems with impulsive effects, J. Math. Anal. Appl. 166 (1992) 563–576. X.Z. Liu, A. Willms, Impulsive stabilizability of autonomous systems, J. Math. Anal. Appl. 187 (1994) 17–39. Sanyi Tang, Lansun Chen, The periodic predator–prey Lotka–Volterra model with impulsive effect, J. Mech. Medicine Biol. 2 (2002) 267–296. Bing Liu, Yujuan Zhang, Lansun Chen, The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management, Nonlinear Anal. RWA 6 (2005) 227–243. L. Stone, B. Shulgin, Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Modelling 31 (2000) 207–215. Alberto d’Onofrio, Pulse vaccination strategy in the SIR epidemic model: Global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Modelling 36 (2002) 473–489. Alberto d’Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biosci. 179 (2002) 57–72. K. Berthier, M. Langlasis, P. Auger, D. Pontier, Dynamics of a feline virus with two transmission models with exponentially growing host populations, Proc. Roy. Soc. Lond. B 267 (2000) 2049–2056.
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[17] M.C.M De Jong, O. Diekmann, J.A.P. Heesterbeek, How does infection depend on the population size? in: D. Mollison (Ed.), Epidemic Models, their Structure and Relation in Data, Cambridge Univ. Press, 1994. [18] W.D. Hamilton, R. Axelrod, R. Tanese, Sexual reproduction as an adaptation to resist parasite (a review), Proc. Natl. Acad. Sci. USA 87 (1990) 3566–3573. [19] A.M. Hammond, T.N. Hardy, Quality of diseased plants as host for insects, in: E.A Heinrichs (Ed.), Plant Strees–Insect Interactions, John-Wiley & Sons, New York, 1989, pp. 381–432. [20] J.C. Holmes, W.M. Bethel, Modifications of intermediate host behaviour by parasite, in: E.V. Canning, C.A Wright (Eds.), Behavioral Aspects of Parasite Transmission, in: Suppl I to Zool. f. Linnean Soc., vol. 51, 1972, pp. 123–149. [21] A. Lakmeeh, O. Arino, Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynam. Contin. Discrete Impulsive Systems 7 (2000) 265–287. [22] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977. [23] D. Greenhalgh, M. Haque, A predator–prey model with disease in prey species only, Math. Methods Appl. Sci. 30 (2007) 911–929. [24] M. Haque, E. Venturino, Increase of the prey may decrease the healthy predator population in presence of a disease in the predator, HERMIS J. 7 (2006) 38–59. [25] Z. Jin, M. Haque, Global stability analysis of an eco-epidemiological model of the Salton sea, J. Biol. Systems 14 (3) (2006) 373–385. [26] E. Venturino, The influence of diseases on Lotka–Volterra systems, Rocky Mountain J. Math. 24 (1994) 381–402. [27] M. Haque, E. Venturino, An eco-epidemiological model with disease in predator: The ratio-dependent case, Math. Methods Appl. Sci. 30 (2007) 1791–1809. [28] M.R. Gonzalez, C.M. Hart, J.R. Verfailile, S.H. Hurlbert, Salinity and fish effects on Salton Sea microecosystems: Water chemistry and nutrient cycling, Hydrobiologia 381 (1988) 105–128.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
An impulsive predator–prey model with communicable disease in the prey species only Maoxing Liu a , Zhen Jin a , Mainul Haque b,∗ a
Department of Mathematics, North University of China, Taiyuan, Shanxi, 030051, PR China
b
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
article
info
Article history: Received 9 December 2007 Accepted 8 October 2008 Keywords: Eco-epidemiology Susceptible prey Infected prey Predator Impulsive effect Bifurcation Coincidence degree
a b s t r a c t A system of impulsive differential equations describing predator–prey dynamics with impulsive effect is proposed and analyzed with the assumption that a transmissible disease is spreading among the prey species only. At first, the ‘‘semi-trivial’’ periodic solution (S (t ), 0, 0) is given. After that, the existence of ‘‘infection-free’’ periodic solution (S (t ), 0, P (t )) and the ‘‘predator-free’’ periodic solution have been obtained via bifurcation. Finally, the method of coincidence degree has been used to derive a set of sufficient conditions for the existence of at least one strictly positive periodic solution. Numerical simulations and a brief discussion conclude the paper. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction After the pioneering work of Vito Volterra and Alfred James Lotka in the middle of twentieth century for predator–prey interactions, the seminal models and competitive system have been studied extensively during the last couple of decades. On the other hand, after the fundamental work of Kermack–McKendrick on SIRS (Susceptible–Infected–Recover–Susceptible) epidemiological models have also received much attention from a group of scientists. But in recent times, these two major fields of study are merged and renamed as a new field of study called ‘‘eco-epidemiology’’. A good number of studies have already been performed in this particular direction (see Hadeler and Freedman [1], Freedman [2], Beltrami and Carroll [3], Xiao and Chen [6,7], Hethcote et al. [8], Haque and his colleagues [4,5,23–27] and the references therein). Impulsive methods have been applied in almost every field of applied sciences. The theoretical investigations and their applications can be found in [9,10]. Recently, the impulsive effect on the ecological situations have been studied by several researchers, for instance, a classical periodic Lotka–Volterra predator–prey system with impulsive effect is investigated by Tang and Chen [11]. Bing Liu and her co-workers have shown the dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management [12]. Similarly, at the same time, the role of impulsive effect on epidemic models has also been investigated; for example, Shulgin et al. have analyzed an SIR epidemic model with pulse vaccination strategy [13]. Alberto d’Onofrio has studied the stability properties of pulse vaccination strategy in SIR and SEIR epidemic models [14,15]. But we know so far there has been no work that dealt with the impulsive effect on the eco-epidemiological system. This is a first attempt to look for the influence of impulsive effect on an eco-epidemiological system. The first eco-epidemiological model has been considered by Hadeler and Freedman [1] in which a disease was spreading among interacting populations. In Venturino [26] the Lotka–Volterra model is taken as the demographic basis to study the influence of a disease propagating in one of the two species. In an aquatic medium, a model of this kind has been first
∗
Corresponding author. Tel.: +44 0 115 951 4949; fax: +44 0 115 951 4951. E-mail address: [email protected] (M. Haque).
1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.010
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introduced in Beltrami and Carroll [3]. Jin and Haque [25] have shown the global stability analysis of an eco-epidemiological model of the Salton sea, the largest lake of California. This lake has been facing problems due to its massive fish and bird mortality. Four types of fish are very common in the Salton Sea, namely Tilapia (Oreochromis mossambicus), Corvina (Cynoscian Xanthulus), Coraker (bairdiella icistius) and Sargo (Anisopremus davidsoni). The actual cause of death is still a mystery, although a toxic algal bloom is being investigated as a possible answer. A transmissible disease is running through the prey species (fish). The fish-eating birds feed on them. At the end of the article, the authors are able to show that for global persistence of the prey and predator (fish-eating birds like ‘‘Pelicans’’) populations from the explosive epidemic in the Salton Sea, reduction of the prey population (especially the infected fishes) in a considerable amount is essential, and hence suitable harvesting policy must be implemented. Their findings are also supported by experimental results [28]. Now in order to apply impulsive effect on this kind of situation one could ask a question; what will happen if people catch these fish impulsively according to seasons? In order to get a proper understanding of this kind of question, we have proposed and analyzed an eco-epidemiological model with impulsive effect. Nevertheless, with the pulse vaccination strategy like [13], one could incorporate the impulsive effect on eco-epidemic situations. Therefore our attempt will open a new window of research in this direction. The article is organized as follows: we have proposed an eco-epidemiological model with impulsive effect in Section 2; the existence and the stability of the ‘‘semi-trivial’’ solution is given in Section 3. In Section 4, we have analyzed the existence and the bifurcation of the ‘‘infection-free’’ solution and the ‘‘predator-free’’ solution, and the existence of a positive periodic solution is analyzed in Section 5. 2. An eco-epidemiological model with impulsive effect 2.1. The basic ecological assumptions (A1) In the absence of transmissible disease the prey population grows according to the logistics law with carrying capacity K (K ∈ R+ ) and intrinsic birth rate constant γ (γ ∈ R+ ). (A2) In the presence of disease, the prey population is divided into two classes, namely susceptible prey, denoted by S (t ), and infected prey, denoted by I (t ). (A3) The result of Bertheir et al. [16] shows that a mass-action incidence assumption is more appropriate than a proportional mixing one in describing the dynamics of direct transmission. Again data of Greenwood [17] experiment suggests that there is no change of qualitative properties upon the contact process whether it follows the law of mass action or proportional mixing. Therefore we assume that the transmission of the disease follow the simple law of mass action λSI like [23,24], where λ is the force of infection. (A4) We assume that only susceptible prey is capable of reproducing and contributing to its carrying capacity. We also assume that the infected prey do not grow and reproduce, but the infected prey can be recovered. The model of Hamilton et al. [18] assumes that infected individuals do not reproduce, infection rather reduces the remaining capacity due to inability to compete for resources. (A5) Nowadays it has been observed that viral, bacterial, fungal parasites can mediate hosts vulnerable to predation [19]. The review of Holmes and Bethel [20] contains many examples in which the parasites change the external features or behavior of the prey, so infected prey may live in the locations that are more accessible to predator. Also, due to infection, they become less active and therefore could be caught easily by the predator compared to susceptible individuals. Thus we assume that searching coefficient of the predator for infected prey is greater than that of susceptible prey. 2.2. The basic differential equations With the above assumptions we get the following set of autonomous non-linear differential equations:
S 0 S = γ S 1 − − c1 SP − λSI + µI , t 6= tn . K 0 I = λSI − c2 IP − µI − δ1 I , tn+1 = tn + T , P 0 = e(c1 SP + c2 IP ) − δ2 P . S (t + ) = (1 − h1 )S (t − ), t = tn . + − I (t +) = (1 −− h2 )I (t ), n = 0, 1, 2 . . . , P (t ) = P (t ).
(2.1)
Here, system (2.1) has to be analyzed with the following initial conditions: S (0) = S0 > 0;
I (0) = I0 > 0;
P (0) = P0 > 0;
(2.2)
where µ is the recover rate, δ1 , the total death rate of the infected prey (the natural death rate of the prey and the death rate of the prey due to infection), δ2 , the death rate of predator population, e, the conversion factor, and c1 and c2 are the searching coefficients of the predator for susceptible and infected prey respectively. We assume that c2 ≥ c1 .
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Let us assume that in a single pulse, the pulse scheme proposes to control a fraction h1 of susceptible prey and a fraction h2 of the infected prey, which is being applied for every T time. We also assume that due to this pulse, the amount of prey leaving from the poll of susceptible and infected individuals are respectively h1 S and h2 I. 3. Existence of the ‘semi-trivial’ periodic solution Let us consider the following single Logistic model with impulsive effect:
S
, t 6= tn , tn+1 = tn + T , K S (t ) = (1 − h1 )S (t − ), t = tn , n = 0, 1, 2 . . . . S0 = γ S
1−
(3.1)
+
It is easy to show that system (3.1) admits a positive periodic solution if and only if T > τ0 = unique and stable if it exists. In fact, for system (3.1), we carry out the change of variable x = linear non-homogeneous impulsive equation
(
− ln(1−h1 ) . γ
Moreover, it is
and obtain the following
γ
, t 6= tn , tn+1 = tn + T , K x(t ) = (1 − h1 )x(t − ), t = tn , n = 0, 1, 2 . . . . x0 = −γ x +
1 , S
+
(3.2)
Let
Y
W (t , s) =
s≤τk
1 1 − h1
exp(−γ (t − s))
(3.3)
be the Cauchy matrix for the respective homogeneous equation. Then x(t ) = W (t , 0)x(0) +
γ
t
Z
K
W (t , s)ds
(3.4)
0
is a solution of system (3.2). This solution is T -periodic if x(0) = x(T ), or if
(1 − W (T , 0))x(0) =
γ K
T
Z
W (t , s)ds.
(3.5)
0
Since the multiplier W (T , 0) of the homogeneous equation x0 = −γ x, t 6= tn , tn+1 = tn + T , x(t + ) = (1 − h1 )x(t − ), t = tn , n = 0, 1, 2 . . . ,
− ln(1−h )
γ
(3.6)
RT
1 is less than 1 because we have γ > , and K 0 W (t , s)ds > 0. Eq. (3.6) has a unique solution x(0) = p0 > 0. T By using the initial condition x(0) = p0 , one could obtain the corresponding unique positive T -periodic solution of system (3.2), and then the unique T -periodic solution of system (3.1) could be derived from there. Obviously this solution is also positive and stable. Thus we have done.
4. Existence of the ‘infection-free’ periodic solution and the ‘predator-free’ solution 4.1. Preliminaries In this section, we followed the technique used by A. Lakmeche and O. Arino in [21], here we gave the existence of the ‘infection-free’ and the ‘predator-free’ periodic solutions. We introduced some notations and stated some preliminary results. Let us consider the following impulsive system given below
0 S (t ) = F1 (S (t ), P (t )), t 6= tn , 0 P (t ) = F2 (S (t ), P (t )), tn+1 = tn + T , + S (t +) = θ1 (S (t ), P (t )), t = tn , P (t ) = θ2 (S (t ), P (t )), n = 0, 1, 2 . . . .
(4.1)
Suppose that
S 0 (t ) = F1 (S (t ), 0), t 6= tn , tn+1 = tn + T , S (t + ) = θ1 (S (t ), 0), t = tn , n = 0, 1, 2 . . .
(4.2)
which is obtained by taking P (t ) = 0 in (4.1), has a periodic solution denoted by U˜ (t ). Denote a trivial periodic solution of system (4.1) by π = (U˜ , 0)T . Letting Φ be the flow associated to system (4.1), we have U (t ) = Φ (t , S0 , P0 ), 0 < t ≤ T ,
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where U0 = U (S0 , P0 ). The flow Φ applies to time T . So, U (T ) = Φ (T , U0 ). The following notations of [21] will be used:
∂θ2 ∂ Φ2 d0 = 1 − (τ0 , U0 ), ∂P ∂P ∂θ1 ∂ Φ1 a00 = 1 − (τ0 , U0 ), ∂S ∂S ∂θ1 ∂ Φ1 b00 = − (τ0 , U0 ), ∂S ∂P ∂ 2 θ2 ∂ Φ1 (τ0 , U0 ) ∂ Φ1 (τ0 , U0 ) 1 ∂θ1 ∂ Φ1 (τ0 , U0 ) ∂ Φ2 (τ0 , U0 ) B = − + ∂ S∂ P ∂τ ∂S a00 ∂ S ∂τ ∂P 2 ∂θ2 ∂ Φ2 (τ0 , U0 ) 1 ∂θ1 ∂ Φ1 (τ0 , U0 ) , − ∂P ∂ S∂ P a00 ∂ S ∂τ 0 ∂ 2 θ2 b0 ∂ Φ1 (τ0 , U0 ) ∂ Φ1 (τ0 , U0 ) ∂ Φ2 (τ0 , U0 ) ∂ 2 θ2 ∂ Φ2 (τ0 , U0 ) 2 C = −2 − 0 − ∂ S∂ P a0 ∂S ∂P ∂P ∂P2 ∂P 0 2 2 ∂θ2 b0 ∂ Φ2 (τ0 , U0 ) ∂θ2 ∂ Φ2 (τ0 , U0 ) − , +2 ∂ P a00 ∂ S∂ P ∂I ∂P2 0
where τ0 is the root of d00 = 0, and
Z t ∂ Φ1 (t , U0 ) ∂ F1 (π (ξ )) = exp dξ , ∂S ∂S 0 Z t ∂ F2 (π (ξ )) ∂ Φ2 (t , U0 ) = exp dξ , ∂P ∂P 0 Z t Z v Z t ∂ Φ1 (t , U0 ) ∂ F1 (π (ξ )) ∂ F1 (π (v)) ∂ F2 (π (ξ )) = exp dξ exp dξ dv, ∂P ∂S ∂P ∂P 0 v 0 Z t 2 Z v Z t 2 ∂ Φ2 (t , U0 ) ∂ F2 (π (ξ )) ∂ F2 (π (v)) ∂ F2 (π (ξ )) = exp dξ exp dξ dv, ∂P∂S ∂P ∂P∂S ∂P 0 v 0 Z t 2 Z v Z t 2 ∂ F2 (π (ξ )) ∂ F2 (π (ξ )) ∂ Φ2 (t , U0 ) ∂ F2 (π (v)) = exp d ξ exp d ξ dv, 2 ∂P2 ∂P ∂P 0 ∂ P2 0 Z t v Z t ∂ F2 (π (ξ )) ∂ F2 (π (v)) + exp dξ ∂ P 0 Z Zv v ∂P∂S Z θ v ∂ F1 (π (ξ )) ∂ F1 (π (v)) ∂ F2 (π (ξ )) × exp dξ exp dξ dθ dv, ∂P ∂P ∂P 0 θ 0 Z t ∂ 2 Φ2 (t , U0 ) ∂ F2 (π (v)) ∂ F2 (π (ξ )) = exp dξ , ∂ P ∂τ ∂P ∂P 0 ∂ Φ1 (τ0 , U0 ) = S˜ 0 (τ0 ), ∂τ then we get the following lemma. Lemma 4.1. If |1 − a00 | < 1 and d00 = 0, then we get: (a) If BC 6= 0, then we have a bifurcation. Moreover, we have a bifurcation of a nontrivial periodic solution of system (4.1) if BC < 0, and a subcritical case if BC > 0. (b) If BC = 0, then we have an undetermined case. 4.2. Existence of the ‘infection-free’ periodic solution Let us consider the following set of differential equations
S S0 = γ S 1 − − c1 SP , F1 (S , P ), K 0 P = ec1 SP − δ2 P , F2 (S , P ), + S h1 )S (t − ) , θ1 (S , P ), (t +) = (1 − − P (t ) = P (t ) , θ2 (S , P ),
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . .
(4.3)
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Fig. 1. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, δ1 = 0.02, δ2 = 0.0139, h1 = 0.3, T = 2.
which satisfy F2 (S , 0) = θ2 (S , 0) ≡ 0, θ1 6= 0 for S 6= 0 and θ2 6= 0 for P 6= 0. In order to apply Lemma 4.1, we compute the following: d00 = 1 − e
R τ0 ∂ F2 (λ(ξ )) ∂I
0
If d0 = 0, we can get τ0 = In addition, 0
dξ
= 1 − e−δ2 τ0 +λ
b00 = −(1 − h1 )
0
s˜(ξ )dξ
.
−ec1 K ln(1−h1 ) . γ (ec1 K −δ2 )
a00 = 1 − (1 − h1 )eγ τ0 − τ0
Z
R τ0
2γ K
0
s˜(ξ )dξ
> 0,
2γ 0 ν γ − K s˜(ξ ) dξ
Rτ
e
R τ0
(−c1 s˜(ν))e(−δ2 τ0 )+ec1
R τ0 0
s˜(ξ )dξ
dν > 0.
0
∂θ
∂θ
∂θ
∂θ
∂2θ
∂2θ
Since ∂ P1 = ∂ S2 = 0, ∂ S1 = 1 − h1 , ∂ P2 = 1, ∂ P 22 = ∂ S ∂ 2P = 0, it is easy to verify that C > 0, and if ec1 then R τ0 (1 − h1 )(1 − h2 )S˜ 0 (τ0 ) ∂ 2 Φ2 (τ0 , U0 ) B=− + [ec1 S˜ (τ0 ) − δ2 ]e 0 · 0 a0 ∂ S∂ I
∂ F2 (λ(ξ )) dξ ∂P
R τ0 0
S˜ (ξ )dξ − δ2 τ0 > 0,
! < 0.
According to Lemma 4.1, we get the following result. Theorem 4.1. System (2.1) has a positive periodic ‘infection-free’ solution if T > τ1 and is close to τ1 , where τ1 satisfies
τ1 =
−ec1 K ln(1 − h1 ) . γ (ec1 K − δ2 )
(4.4)
This theorem illustrates that the ‘semi-trivial’ solution becomes unstable if the period T is more than τ1 and close to τ1 , and become the ‘infection-free’ solution. Numerical simulations confirm our analytical results (Figs. 1 and 2). 4.3. Existence of the ‘predator-free’ periodic solution Let us consider the following set of differential equations
γ 2 ˙ S = γ S − K S − λSI + µI , F1 (S , I ), ˙I = −δ1 I + λSI − µI , F2 (S , I ), S (t + ) = (1 − h1 )S (t − ) , θ1 (S , I ), + I (t ) = (1 − h2 )I (t − ) , θ2 (S , I ),
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . .
with F2 (S , 0) = θ2 (S , 0) ≡ 0, θ1 6= 0 for S 6= 0 and θ2 6= 0 for I 6= 0. We compute the following: d00 = 1 − e
R τ0 ∂ F2 (λ(ξ )) 0
∂I
If d0 = 0, we can get τ0 = 0
dξ
= 1 − e(−(δ1 +µ)τ0 )+λ
−λK ln(1−h1 ) . γ (λK −δ1 −µ)
R τ0 0
s˜(ξ )dξ
.
(4.5)
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Fig. 2. Time-series for the P (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, δ2 = 0.0139, h1 = 0.3, T = 2.
In addition, a00 = 1 − (1 − h1 )eγ τ0 − b00 = −(1 − h1 )
τ0
Z
e
2γ K
R τ0 0
s˜(ξ )dξ
> 0,
Rτ 2γ 0 ν γ − K s˜(ξ ) dξ
(−λ˜s(ν) + µ)e(−(δ1 +µ)τ0 )+λ
R τ0 0
s˜(ξ )dξ
dν > 0 .
0
∂θ
∂θ
∂θ
∂θ
∂2θ
∂2θ
Since ∂ I1 = ∂ S2 = 0, ∂ S1 = 1−h1 , ∂ I2 = 1−h2 , ∂ I 22 = ∂ SI2 = 0, it is easy to verify that C > 0, and if λ then R τ0 (1 − h1 )(1 − h2 )S˜ 0 (τ0 ) ∂ 2 Φ2 (τ0 , U0 ) · + [λS˜ (τ0 ) − δ1 ]e 0 B=− 0 a0 ∂ S∂ I
∂ F2 (λ(ξ )) dξ ∂I
R τ0 0
S˜ (ξ )dξ −δ1 τ0 < 0,
! < 0.
According to Lemma 4.1, we get the following result. Theorem 4.2. System (2.1) has a positive periodic ‘predator-free’ solution if T > τ2 and is close to τ2 , where τ2 satisfies
τ2 =
−λK ln(1 − h1 ) . γ (λK − δ1 − µ)
(4.6)
This theorem illustrates that the ‘semi-trivial’ solution becomes unstable if the period T is more than τ2 and close to τ2 , the variable I (t ) begins to oscillate with an amplitude, and become the ‘predator-free’ solution. Numerical simulations confirm our theoretical results (Figs. 3 and 4). 5. Existence of the positive T -periodic solution We assume µ = 0 through out this section. Before using the method of coincidence degree, we shall mention some definitions and basic concepts from the book of Gaines and Mawhin [22], interested readers are requested to refer the book. Let X , Z be real Banach spaces, L : Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z . If L is a Fredholm mapping of index zero there exist continuous projects P : X → X and Q : Z → Z such that Im P = Ker L, Ker Q = Im L = Im(I − Q ). It follows that L |Dom L T Ker P : (I − P )X → Im L is invertible. We denote the inverse of that
¯ if QN (Ω ¯ ) is bounded and mapping by Kp . If Ω is an open bounded subset of X , the mapping N will be called L-compact on Ω ¯ Kp (I − Q )N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exit isomorphisms J : Im Q → Ker L. In the proof of the existence theorem below, we will use the continuation theorem from Gaines and Mawhin [22]. ¯ . Suppose Lemma 5.1. Let L be a Fredholm mapping of index zero and let N be L-compact on Ω ¯ Ω. (a) For each β ∈ (0, 1), every solution x of Lx = β Nx is such that x∈∂ (b) QNx = 6 0 for each x ∈ ∂ Ω ∩ Ker L and deg{JQN , Ω ∩ Ker L, 0} 6= 0. ¯. Then the equation Lx = Nx has at least one solution which belongs to Dom L ∩ Ω
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Fig. 3. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, λ = 0.4, u = 0.01, δ1 = 0.02, h1 = 0.3, h2 = 0.1, T = 1.
Fig. 4. Time-series for the I (t ) evolving according to the model with γ = 0.5, K = 10, λ = 0.4, u = 0.01, δ1 = 0.02, h1 = 0.3, h2 = 0.1, T = 1.
Theorem 5.1. If γ T + ln(1 − h1 ) > 0 and
λδ2 K > 0, γ T γ c2 T γ ec2 ln(1 − h2 ) δ2 c1 K ln(1 − h2 ) Kc 2 λδ2 c1 K − γ+ + 12 δ2 − + > 0, ec2 c2 γ T T γ c2 γ ec22 ln(1 − h2 ) λK ln(1 − h1 ) λKc1 ln(1 − h2 ) λ2 δ2 K −δ1 + + γ+ + δ − − > 0, 2 2 T c2 γ T T γ c2 γ ec22 K
γ+
ln(1 − h1 )
+
Kc1
δ2 −
ln(1 − h1 )
−
then system (2.1) has at least one positive T -periodic solution. Proof. Making the change of variable S (t ) = es(t ) ,
I (t ) = ei(t ) ,
P (t ) = ep(t )
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then system (2.1) is transformed into,
γ s˙(t ) = γ − es(t ) − λei(t ) − c1 ep(t ) , t 6= tn . K s(t ) p(t ) ˙i(t ) = −δ1 + λe − c2 e , tn+1 = tn + T , s(t ) p˙ (t ) = −δ2 + e(c1 e + ec2 ei(t ) ). + − s ( t ) − s ( t ) = ln ( 1 − h1 ), t = tn . + − i ( t ) − i ( t ) = ln ( 1 − h n = 0, 1, 2 . . . , 2 ), + p(t ) − p(t − ) = 0.
(5.1)
Let Dom L = PCT1 × PCT1 and s L : Dom L → Z , i p
! ∆s(tk ) ∆i(tk ) . ∆p(tk )
s˙ → ˙i , p˙
!
(5.2)
Again assume Dom N = PCT1 × PCT1 → Z with
N
s i p
!
γ−
γ
es(t ) − λei(t ) − c1 ep(t )
!
K −δ1 + λes(t ) − c2 ep(t ) , −δ2 + e(c1 es(t ) ) + ec2 ei(t )
→
ln(1 − h1 ) ln(1 − h2 ) . 0
(5.3)
Obviously,
( Ker L =
s i p
s i p
! :
! =
C1 C2 C3
!
) ∈ R , t ∈ [0, T ] 3
(5.4)
and
Im L =
Z
T
0 ! ! ! l ak Z T m , bk ∈Z : m + bk T = 0 . Z0 n ck T n + ck T = 0 l + ak T = 0
z=
(5.5)
0
Since Im L is closed in Z , L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ),
where
Z s i p
!
P
T
l + ak T = 0
0 Z T = m + bk T = 0 , T 0 Z T n + ck T = 0 1
(5.6)
0
and
Z l ak m , bk n ck
!
QZ = Q
!!
1 = T
T
l + ak T = 0
0 Z T 0! m + bk T = 0 , 0 . 0 0 Z T n + ck T = 0 0
(5.7)
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Furthermore, the generalized inverse Kp : Im L → Ker P ∩ Dom L is given by
Z
t
l + ak − Z t0 Kp Z = m + bk − Z0 t n + ck − 0
1
T
Z
t
Z
T Z0 Z0 T t 1
l − ak
m − bk . T Z0 Z0 T t 1 n − ck T
0
(5.8)
0
Thus
Z T Z Z γ s(t ) 1 T i(t ) 1 T 1 1 p(t ) λ e − c e + e − ln ( 1 − h ) γ − 1 1 ! T 0 K T 0! Z T T 0 Z T T 0 s 1 1 1 , 0 , QN i = λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ1 + T 0 0 p Z T T 0 Z TT 1 1 −δ2 + e(c1 es(t ) ) + ec2 ei(t ) T
T
0
(5.9)
0
and
s i p
!
Kp (I − Q )N
=
−
γ
Z t λei(t ) − c1 ep(t ) + ln(1 − h1 ) 0 0 K Z Z t 0 t −δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) 0 0 Z t Z t s(t ) i(t ) −δ2 + e(c1 e ) + ec2 e 0 0 1 Z t h i γ s(t ) γ − e − λei(t ) − c1 ep(t ) + ln(1 − h1 ) T 0 Z K t 1 [−δ1 + λes(t ) − c2 ep(t ) ] + ln(1 − h2 ) T 0 Z t 1 s(t ) i(t ) [−δ2 + ec1 e + ec2 e ]
γ−
Z
t
es(t ) −
T
Z
t
0
Z t h i 1 γ − γ − es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 ) T 2 K 0 Z t 1 1 − [−δ1 + λes(t ) − c2 ep(t ) ] + ln(1 − h2 ) T 2 0 Z t 1 1 − [−δ2 + ec1 es(t ) + ec2 ei(t ) ]
1 −
T
2
.
(5.10)
0
¯) Clearly, QN and Kp (I − Q )N are continuous. Using the Arzela–Ascoli Theorem, it is not difficult to show that Kp (I − Q )N (Ω is compact for any open bounded set Ω ⊂ X . The isomorphism J of Im Q onto Ker L may be defined by J : Im Q → X ,
h1 h2
,
0 0
→
h1 h2
.
(5.11)
Now corresponding to the operator equation s L i p
!
s i , p
!
= βN
(5.12)
we have
h i γ s˙(t ) = β γ − es(t ) − λei(t ) − c1 ep(t ) , K s(t ) p(t ) ˙i(t ) = β[−δ1 + λe − c2 e ], s(t ) p˙ (t ) = β[−δ2 + e(c1 e ) + ec2 ei(t ) ], + − s(t+ ) − s(t− ) = β ln(1 − h1 ), i(t +) − i(t −) = β ln(1 − h2 ), p(t ) − p(t ) = 0.
t 6= tn , tn+1 = tn + T , t = tn , n = 0, 1, 2 . . . ,
(5.13)
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We prove the existence of M0 such that every T -periodic solution of system (2.1) satisfies |s(t )| + |i(t )| + |p(t )| ≤ M0 , let (s(t ), i(t ), p(t )) be a T -periodic solution. By integrating (5.13) we obtain
Z T h i γ s(t ) e + λei(t ) + c1 ep(t ) dt = γ T + ln(1 − h1 ), K Z0 T [λes(t ) − c2 ep(t ) ]dt = δ1 T − ln(1 − h2 ), 0 Z T [ec1 es(t ) + ec2 ei(t ) ]dt = δ2 T .
(5.14)
0
Since s(t ), i(t ), p(t ) ∈ X , there exist ξ¯ : s(ξ¯ ) = maxt ∈[0,T ] s(t ), η¯ : i(η) ¯ = maxt ∈[0,T ] i(t ), ζ¯ : p(ζ¯ ) = maxt ∈[0,T ] p(t ). On the other hand, there also exist ξ , η, ζ , such that: s(ξ + ) = inft ∈[0,T ] s(t ), i(η+ ) = inft ∈[0,T ] i(t ), p(ζ + ) = inft ∈[0,T ] p(t ). Here, if ξ , η, ζ is not an impulsive point, then we have s(ξ + ) = s(ξ ), i(η+ ) = i(η), p(ζ + ) = p(ζ ); if it is an impulsive point tk , then we have s(ξ + ) = s(tk+ ), i(η+ ) = i(tk+ ), p(ζ + ) = p(tk+ ). It follows from system (5.14) that
Z T |˙s(t )| ≤ 2(γ T + ln(1 − h1 )), Z0 T ¯ ¯ |˙i(t )| ≤ δ1 T + λT es(ξ ) + c2 eT es(ζ ) + ln(1 − h2 ), Z0 T |˙p(t )| ≤ 2δ2 T .
(5.15)
0
From (5.13) it follows that:
γ K
es(ξ ) + λei(η) + c1 e
p(ζ )
ln(1 − h1 )
≤γ +
T
,
(5.16)
then we can get
K ln(1 − h1 ) , s(ξ ) ≤ ln γ γ + T 1 ln(1 − h1 ) i(η) ≤ ln γ+ , λ T p(ζ ) ≤ ln 1 γ + ln(1 − h1 ) . c1
(5.17)
T
So s(t ) ≤ s(ξ ) +
T
Z
|˙s(t )| ≤ ln
0
K
γ
γ+
ln(1 − h1 )
T
+ 2(γ T + ln(1 − h1 )) , sM ,
ln(1 − h1 ) 1 ¯ ¯ |˙i(t )| ≤ ln γ+ + δ1 T + λT es(ξ ) + c2 eT es(ζ ) + ln(1 − h2 ) , iM , λ T 0 Z T 1 ln(1 − h1 ) p(t ) ≤ p(ζ ) + |˙p(t )| ≤ ln γ+ + 2δ2 T , pM . i(t ) ≤ i(η) +
Z
T
c1
0
T
We shall give here lower bound for sL , iL , pL as follows. From (5.14) we get
Z T λδ2 T γ c1 es(t ) dt = γ T + ln(1 − h1 ) + (δ2 T − ln(1 − h2 )) − , K c ec2 2 0 Z T Z T s(t ) λ e dt − c ep(t ) dt = δ1 T − ln(1 − h2 ), 2 0Z 0Z T T ec1 es(t ) dt + ec2 ei(t ) dt = δ2 T . 0
(5.18)
0
In fact, we obtain from the first equation of (5.18) s(ξ¯ ) ≥ ln
K
γ
γ+
ln(1 − h1 ) T
+
Kc1
γ c2
δ2 −
ln(1 − h1 ) T
−
λδ2 K , γ ec2
(5.19)
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then we get s(t ) ≥ s(ξ¯ ) −
T
Z
|˙s(t )|dt ≥ ln
K
γ+
ln(1 − h1 )
+
ln(1 − h1 ) λδ2 K δ2 − − γ c2 T γ ec2 Kc1
γ T 0 − 2(γ T + ln(1 − h1 )) , sL , Z T δ2 c1 K ln(1 − h2 ) Kc12 ln(1 − h2 ) λδ2 c1 K ˙ |i(t )|dt ≥ ln i(t ) ≥ i(η) ¯ − − (γ + ) + 2 δ2 − + ec2 c2 γ T T γ c2 γ ec22 0 s(ξ¯ ) s(ζ¯ ) − (δ1 T + λT e + c2 eT e + ln(1 − h2 )) , iL , Z T ln(1 − h2 ) λK ln(1 − h1 ) |˙p(t )|dt ≥ ln −δ1 + p(t ) ≥ p(ζ¯ ) − + γ+ T c2 γ T 0 λ2 δ2 K λKc1 ln(1 − h2 ) − + δ2 − − 2δ2 T , pL , T γ c22 γ ec22 therefore we obtain max |s(t )| ≤ max{|sM |, |sL |} , Ms ,
t ∈[0,T ]
max |i(t )| ≤ max{|iM |, |iL |} , Mi ,
t ∈[0,T ]
max |p(t )| ≤ max{|pM |, |pL |} , Mp .
t ∈[0,T ]
Clearly, Ms , Mi , Mp are independent of β , and M0 = Ms + Mi + Mp . By Lemma 5.1 and the assumptions in Theorem 5.1, it is not difficult to show that the system of algebraic equations
γ u + λv + c1 w = γ T + ln(1 − h1 ), K λu − c2 w = δ1 T − ln(1 − h2 ), ec u + ec v = δ T 1 2 2
(5.20)
has a unique solution (u∗ , v ∗ , w ∗ ) ∈ R3 . Let B be a ball of R3 centered at the origin of radius greater than γ T + ln(1 − h1 ), and
Ω = {X = (s, i, p) ∈ PCT1 × PCT1 × PCT1 : kxk < B}. It is clear that Ω verifies the requirement (a) in Lemma 5.1, when X ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R3 , X is a constant vector in R3 with kX k = B. Then
γ−
γ K
es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 )
,
−δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ2 + ec1 es(t ) + ec2 ei(t )
QNX =
0 0 0
!
and for X ∈ Ker L ∩ Ω we have
JQNX =
γ−
γ K
es(t ) − λei(t ) − c1 ep(t ) + ln(1 − h1 )
−δ1 + λes(t ) − c2 ep(t ) + ln(1 − h2 ) −δ2 + ec1 es(t ) + ec2 ei(t )
.
Furthermore, in view of assumption in Theorem 5.1, it is to easy to prove that
γ
deg {JQNX , Ω ∩ Ker L, 0} = sign det K λ ec1
λ 0 ec2
c1
∗ ∗ ∗ −c2 u v w 6= 0. 0
Therefore we have already proved that Ω satisfied all the requirements of Theorem 5.1. Hence, system (2.1) has at least ¯ , which implies that the result of Theorem 5.1 is true and thus the proof is complete. To one positive periodic solution in Ω substantiate our analytical finding we have performed the numerical simulations (Figs. 5–7).
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Fig. 5. Time-series of the S (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
Fig. 6. Time-series for the I (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
6. Discussion In this paper, we have proposed and analyzed a Lotka–Volterra type predator–prey system with disease in prey species with impulsive effect. A very similar model with out impulsive effect has been considered by Beltrami and Carroll [3] to present a phytoplankton–zooplankton system. They have compared their results with experimental results. Therefore we claim that our model is very much practical and widely used (as it is Lotka–Volterra type) in the field of ‘‘eco-epidemiology’’. At first, the ‘semi-trivial’ periodic solution (S (t ), 0, 0) has been shown, and then the existence of ‘infection-free’ periodic solution (S (t ), 0, P (t )) and ‘predator-free’ periodic solution near the ‘semi-trivial’ periodic solution are analyzed via bifurcation. Finally, by using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive periodic solution. From our analysis it can be observed that if the time period T is more than τ1 and close to τ1 , then impulsive ecoepidemiological model has the ‘infection-free’ solution. This is an important result from ecological view point since in this case only sound prey and the predator species co-exist and the transmissible disease dies out from the eco-system. So we have found out the range of time period under which the eco-system could be made disease-free. On the other hand, if the time period T is more than τ2 and close to τ2 , then susceptible prey and the infective prey are stable co-existent, that is under this conditions predator populations extinct. Therefore, considering both the situations, we can conclude that our model is able to search the value of time period T for which the predator population extinct and when it persist or in other
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Fig. 7. Time-series of the P (t ) evolving according to the model with γ = 0.5, K = 10, e = 0.9, c1 = 0.02, c2 = 0.029, δ1 = 0.02, δ2 = 0.01393, h1 = 0.3, h1 = 0.3, T = 1.
words, when the system will have infection and when it will be infection free. We would also like to mention that we have derived the conditions for which the solution of system (4.1) goes to a Hopf-bifurcation situation and also found out the conditions for which it is subcritical (Theorem 4.1). Finally, a set of sufficient conditions are obtained for the existence of at least one strictly positive periodic solution by using the method of coincidence degree. Note that in this case the three species, namely, sound prey, infected prey and the predator population exist in the eco-system. Our analytical results are confirmed by our numerical simulations. Before ending this article we would like to mention that this work is the first attempt to show the effect of impulse on eco-epidemiological model. So there is a room of improvement. We mention some future directions of research based on this article: (i) One may consider the situation where the predator is infected by eating infected prey. There should either be distinction made for healthy and infected predator, to account for the difference in mortality, or having the predator mortality dependent on the time-varying density of the infected prey. (ii) The time delay or latent period between contact, between susceptible and infected prey and the susceptible prey becoming infected could also be modelled and (iii) the infection rate may also be taken as ratio-dependent. These are the possible directions for future research. Acknowledgments The author Mainul Haque would like to thank to his teacher Professor John R. King, associate editor of SIAM J. Appl. Math., for his continuous encouragement and suggestions. A crucial part of this work was done when the third author Mainul Haque visited the North University of China, PR China, with the financial support of National Natural Science Foundation of China (10471040). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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