Citation: Safa Sadaghiani, Soghra Fallahi, Hossein Heshmati, Saeed Hosseini Teshnizi, Hooshang Abedini Chaijan, Fahimeh Farzad Amir Ebrahimi, Farhad Khorrami, Mina Poorrezaeian, Faezeh Alizadeh. Effect of antioxidant supplements on sperm parameters in infertile male smokers: a single-blinded clinical trial[J]. AIMS Public Health, 2020, 7(1): 92-99. doi: 10.3934/publichealth.2020009
[1] | Parvaiz Ahmad Naik, Muhammad Amer, Rizwan Ahmed, Sania Qureshi, Zhengxin Huang . Stability and bifurcation analysis of a discrete predator-prey system of Ricker type with refuge effect. Mathematical Biosciences and Engineering, 2024, 21(3): 4554-4586. doi: 10.3934/mbe.2024201 |
[2] | Yuxuan Zhang, Xinmiao Rong, Jimin Zhang . A diffusive predator-prey system with prey refuge and predator cannibalism. Mathematical Biosciences and Engineering, 2019, 16(3): 1445-1470. doi: 10.3934/mbe.2019070 |
[3] | Xiaoyuan Chang, Junjie Wei . Stability and Hopf bifurcation in a diffusivepredator-prey system incorporating a prey refuge. Mathematical Biosciences and Engineering, 2013, 10(4): 979-996. doi: 10.3934/mbe.2013.10.979 |
[4] | Christian Cortés García, Jasmidt Vera Cuenca . Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response. Mathematical Biosciences and Engineering, 2023, 20(8): 13681-13703. doi: 10.3934/mbe.2023610 |
[5] | Jinxing Zhao, Yuanfu Shao . Bifurcations of a prey-predator system with fear, refuge and additional food. Mathematical Biosciences and Engineering, 2023, 20(2): 3700-3720. doi: 10.3934/mbe.2023173 |
[6] | Christian Cortés García . Bifurcations in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and constant prey refuge at low density. Mathematical Biosciences and Engineering, 2022, 19(12): 14029-14055. doi: 10.3934/mbe.2022653 |
[7] | Yuanfu Shao . Bifurcations of a delayed predator-prey system with fear, refuge for prey and additional food for predator. Mathematical Biosciences and Engineering, 2023, 20(4): 7429-7452. doi: 10.3934/mbe.2023322 |
[8] | Rajalakshmi Manoharan, Reenu Rani, Ali Moussaoui . Predator-prey dynamics with refuge, alternate food, and harvesting strategies in a patchy habitat. Mathematical Biosciences and Engineering, 2025, 22(4): 810-845. doi: 10.3934/mbe.2025029 |
[9] | Shuqi Zhai, Qinglong Wang, Ting Yu . Fuzzy optimal harvesting of a prey-predator model in the presence of toxicity with prey refuge under imprecise parameters. Mathematical Biosciences and Engineering, 2022, 19(12): 11983-12012. doi: 10.3934/mbe.2022558 |
[10] | Tingting Ma, Xinzhu Meng . Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism. Mathematical Biosciences and Engineering, 2022, 19(6): 6040-6071. doi: 10.3934/mbe.2022282 |
From the initial work by Lotka [1] and Volterra [2], predator-prey model has become and will continue to be one of the main themes in mathematical biology. In the interaction between predator and prey, the phenomenon of prey refuge always exists. It is assumed that prey species can live in two different regions. One is the prey refuge and the other is the predatory region. From biological view, prey refuge can exists and there are no predators in the prey refuge, so it can help increase the population density of the prey. Further, refuge is an effective strategy for reducing predation as a prey population evolved. For this reason, Gause and his partners [3,4] proposed the predator-prey model with a refuge. Moreover, Magalh˜aes et al. [5] studied the dynamics of thrips prey and their mite predators in a refuge, and they predicted the small effect of the refuge on the density of prey under the equilibrium state. Ghosh et al. [6] investigated the impact of additional food for predator on the dynamics of the predator-prey model with a prey refuge. When in a high-prey refuge ecological system, it was observed that the predator extinction possibility may be removed by supplying additional food [7,8] to predator population. Ufuktepe [9] investigated the stability of a prey refuge predator-prey model with Allee effects. Fractional-order factor was introduced into prey-predator model with prey refuge in Xie [10]. On the other hand, Holling [11] argued that the functional response is an important factor to affect the predator-prey model. The predator may reduce its feeding rate when it is fully saturated, and the feeding rate no longer varies with the increase in prey density. Thus he proposed three types of Holling functional responses. Among them, most of the researchers showed their interest in Holling type Ⅱ functional response [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. For the above reasons, Jana [26] considered the following predator-prey system incorporating a prey refuge:
{dxdt=r1x(1−xk1)−σ1x+σ2y,dydt=r2y(1−yk2)+σ1x−σ2y−αyza+y,dzdt=βy(t−τ)z(t−τ)a+y(t−τ)−dz−γz2, | (1.1) |
where x and y denote the density of the prey in the refuge and in the predatory region at any time t, r1 and r2 denote intrinsic growth rate respectively for the prey population x and y at any time t. Further, the environment carrying capacity are denoted by k1 and k2 respectively for the prey x and y. Then, at any time t, σ1 denotes the per unit migration of the prey in the refuge to the predatory region and σ2 denotes from the predatory region to the refuge. Next, z denotes density of the predator in the predatory region at any time t. In addition, the predator consumes the prey at Holling type Ⅱ functional response αya+y, where α is the maximal predator per capita consumption rate and a is the half capturing saturation constant. Furthermore, d is the natural death rate of predator at any time t, γ is the density dependent mortality rate of predator. And β is the rate of the predator consumes prey (assume that 0<β≤α). Because the reproduction of predators after predating the prey is not instantaneous, we assumed that the time interval between the prey are killed and the corresponding increase in the number of predators are thought to be time delayed τ of the system (1.1).
In the real world application, some authors argued that predators living in the predatory region are classified by two fixed ages [27,28,29,30,31], one is immature predator and the other is mature predator, the immature predator have no ability to attack prey.
Motivated by the above mentioned works, in the present paper, we investigate the periodic solution of the following delayed model:
{dxdt=r1(t)x(t)(1−x(t)k1(t))−σ1(t)x(t)+σ2(t)y(t),dydt=r2(t)y(t)(1−y(t)k2(t))+σ1(t)x(t)−σ2(t)y(t)−α(t)y(t)z2(t)a+y(t),dz1dt=β(t)y(t)z2(t)a+y(t)−β(t−τ)y(t−τ)z2(t−τ)a+y(t−τ)−d1(t)z1(t),dz2dt=β(t−τ)y(t−τ)z2(t−τ)a+y(t−τ)−d2(t)z2(t), | (1.2) |
where z1(t) and z2(t) denote the density of immature predator and mature predator respectively at any time t. Next, r1(t),r2(t),k1(t),k2(t),σ1(t),σ2(t),α(t),β(t),d1(t)andd2(t) are continuously positive periodic functions with period ω. Moreover, d1(t)andd2(t) are the death rate of predator at any time t. The nomenclature y(t−τ)z2(t−τ)a+y(t−τ) stands for the number of immature predator that were born at time (t−τ) which still survive at time t and become mature predator. The initial conditions for the system (1.2) are
(x(t),y(t),z1(t),z2(t))∈C+=C([−τ,0],R4+),x(0)>0,y(0)>0,z1(0)>0,z2(0)>0. |
The aim of this paper is to obtain some sufficient conditions for the existence of positive periodic solution of system (1.2). However, we encounter with some difficulties when we use Mawhin's coincidence degree theory to obtain the periodic solutions. Firstly, the forth equation of system (1.2) has a term y(t−τ)z2(t−τ), rather than y(t−τ)z2(t). If we follow the skill in [32], it will lead us to u3(ξ3)≤ˉα−1(a+l+)[(ˉr2−ˉσ2)+l−1−ˉσ1eu1(η1)], which contains u1(η1). Consequently, we can not get the bound of u3 or u1. We will overcome this difficulty in the following paper.
The rest of our paper is organized as follows: section 2 is to prove the existence of the positive periodic solution of system (1.2). In section 3, an example is to demonstrate the obtained result.
It is not difficult to see that we can separate the third equation of (1.2) from the whole system and obtain the following subsystem of (1.2):
{dxdt=r1(t)x(t)(1−x(t)k1(t))−σ1(t)x(t)+σ2(t)y(t),dydt=r2(t)y(t)(1−y(t)k2(t))+σ1(t)x(t)−σ2(t)y(t)−α(t)y(t)z(t)a+y(t),dz2dt=β(t−τ)y(t−τ)z2(t−τ)a+y(t−τ)−d2(t)z2(t). | (2.1) |
The initial conditions for the system (2.1) are
(x(t),y(t),z2(t))∈C+=C([−τ,0],R3+),x(0)>0,y(0)>0,z2(0)>0. |
In order to obtain the positive periodic solution of (2.1), we need some known and preliminary results.
Lemma 2.1. Let Ω∈U be an open bounded set. Let L be a Fredholm operator of index zero and let N be L-compact on ˉΩ. Suppose that the following conditions are satisfied:
(a) for each λ∈(0,1),u∈∂Ω∩DomL,Lu≠λNu;
(b) for each u∈∂Ω∩kerL,QNu≠0;
(c) deg[JQN,Ω∩kerL,0]≠0.
Then Lu=Nu has at least one solution in ˉΩ∩DomL.
For the notations, concepts and further details of Lemma2.1, one can refer to [33,34,35,36].
Lemma 2.2. If f(t) is a continuously periodic function with period ω, then
∫t+ωωf(s)ds=∫t0f(s)ds,forany t. |
Proof. Let j=s−ω, then
∫t+ωωf(s)ds=∫t0f(j+ω)dj=∫t0f(j)dj=∫t0f(s)ds. |
Lemma 2.3. [37] If u(t) is a continuously differentiable periodic function with period ω, then there is a ˜t∈[0,ω] such that
|u(t)|≤|u(˜t)|+∫ω0|˙u(s)|dsor|u(t)|≥|u(˜t)|−∫ω0|˙u(s)|ds. |
For convenience, we adopt the following notations in our paper:
ˉf=1ω∫ω0f(t)dt,fL=mint∈[0,ω]f(t),fM=maxt∈[0,ω]f(t),
l−=adL2βMe2ˉσ2ω−dL2,l+=adM2βLe−2ˉσ2ω−dM2,u0=adL2ˉβ−a,
b1=12(¯k1r1){(ˉr1−ˉσ1)+[(ˉr1−ˉσ1)2+4ˉσ2l+(¯r1k1)]12},b2=(¯k1r1)(ˉr1−ˉσ1),
b3=ˉα−1a[(ˉr2−ˉσ2)−(¯r2k2)l+],b4=ˉα−1(a+l+)[(ˉr2−ˉσ2)+ˉσ2l−eB1],
b5=12(¯k1r1){(ˉr1−ˉσ1)+[(ˉr1−ˉσ1)2+4ˉσ2u0(¯r1k1)]12},
b6=ˉα−1(a+u0)[(ˉr2−ˉσ2)−(¯r1k1)u0+ˉσ2D6u0],
B1=max{|B11|,|B12|},B2=max{|B21|,|B22|},B3=max{|B31|,|B32|}.
And we assume that:
(H1):dL2≤βMe2ˉσ2ω;
(H2):βL≤(dL2)−1dM2βM;
(H3):ˉr2<ˉαa−1eB3+(¯r2k2)l++ˉσ2.
Now, we are in a position to state our main results.
Theorem 2.1. If system (1.2) satisfies: (H1), (H2) and (H3), then it has at least one positive periodic solution.
Proof. We prove this theorem for two steps.
Step 1: We prove subsystem (2.1) has at least one periodic solution.
Letting
u1(t)=lnx(t),u2(t)=lny(t),u3(t)=lnz2(t), |
then we have
{˙u1(t)=r1(t)(1−eu1(t)k1(t))+σ2(t)eu2(t)−u1(t)−σ1(t),˙u2(t)=r2(t)(1−eu2(t)k2(t))+σ1(t)eu1(t)−u2(t)−α(t)eu3(t)a+eu2(t)−σ2(t),˙u3(t)=β(t−τ)eu2(t−τ)+u3(t−τ)−u3(t)a+eu2(t−τ)−d2(t). | (2.2) |
Take
U=V={u=(u1,u2,u3)∈C(R,R3)|u(t+ω)=u(t)}. |
It is easy to see that U,V are both Banach Spaces with the norm ||⋅||,
||u||=maxt∈[0,ω]|u1|+maxt∈[0,ω]|u2|+maxt∈[0,ω]|u3|,u=(u1,u2,u3)∈UorV.
For any u=(u1,u2,u3)∈U, by the periodicity of the coefficients of system (2.2). We can check that:
r1(t)(1−eu1(t)k1(t))+σ2(t)eu2(t)−u1(t)−σ1(t):=Θ1(u,t),
r2(t)(1−eu2(t)k2(t))+σ1(t)eu1(t)−u2(t)−α(t)eu3(t)a+eu2(t)−σ2(t):=Θ2(u,t)
and
β(t−τ)eu2(t−τ)+u3(t−τ)−u3(t)a+eu2(t−τ)−d2(t):=Θ3(u,t)
are all ω-periodic functions.
In fact,
Θ1(u(t+ω),t+ω)=r1(t+ω)(1−eu1(t+ω)k1(t+ω))+σ2(t+ω)eu2(t+ω)−u1(t+ω)−σ1(t+ω)=r1(t)(1−eu1(t)k1(t))+σ2(t)eu2(t)−u1(t)−σ1(t)=Θ1(u,t). |
In a similar way, one can obtain
Θ2(u(t+ω),t+ω)=Θ2(u,t),Θ3(u(t+ω),t+ω)=Θ3(u,t).
Set
L:DomL⋂UL(u1(t),u2(t),u3(t))=(du1(t)dt,du2(t)dt,du3(t)dt),
where DomL={(u1,u2,u3)∈C(R,R3)} and N:U→U is defined by
N(u1u2u3)=(Θ1(u,t)Θ2(u,t)Θ3(u,t)),
Define
P(u1u2u3)=Q(u1u2u3)=(1ω∫ω0u1(t)dt1ω∫ω0u2(t)dt1ω∫ω0u3(t)dt), (u1u2u3)∈U=V.
It is not difficult to know that
kerL={u∈U|u=C0,C0∈R3} and ImL={v∈V|∫ω0v(t)dt=0}.
Consequently, dimkerL=codimImL=3<+∞, and P and Q are continuous projectors such that
ImP=kerL,kerQ=ImL=Im(I−Q).
It follows that L is a Fredholm mapping of index zero. Moreover, the generalized inverse(of L) Kp:ImL→DomL∩kerP exists and is given by
Kp(v)=∫t0v(s)ds−1ω∫ω0∫t0v(s)dsdt,
then,
QNu=(1ω∫ω0Θ1(u,t)dt1ω∫ω0Θ2(u,t)dt1ω∫ω0Θ3(u,t)dt)
and
Kp(I−Q)Nu=∫t0Nu(s)ds−1ω∫ω0∫t0Nu(s)dsdt−(tω−12)∫ω0Nu(s)ds.
Obviously, QN and Kp(I−Q)N are continuous. By using the Arzela-Ascoli theorem, it's not difficult to see that N is L-compact on ˉΩ with any open bounded set Ω⊂U.
Next, our aim is to search for an appropriate open bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lu=λNu,λ∈(0,1), we have
˙u1(t)=λ[r1(t)(1−eu1(t)k1(t))+σ2(t)eu2(t)−u1(t)−σ1(t)], | (2.3) |
˙u2(t)=λ[r2(t)(1−eu2(t)k2(t))+σ1(t)eu1(t)−u2(t)−α(t)eu3(t)a+eu2(t)−σ2(t)], | (2.4) |
˙u3(t)=λ[β(t−τ)eu2(t−τ)+u3(t−τ)−u3(t)a+eu2(t−τ)−d2(t)]. | (2.5) |
Suppose u=(u1(t),u2(t),u3(t))T∈U is a solution of (2.3), (2.4) and (2.5), for a certain λ∈(0,1). Integrating (2.3), (2.4) and (2.5) over the interval [0,ω], we obtain
ˉσ1ω=∫ω0[σ2(t)eu2(t)−u1(t)+r1(t)−r1(t)k1(t)eu1(t)]dt, | (2.6) |
ˉσ2ω=∫ω0[σ1(t)eu1(t)−u2(t)−α(t)eu3(t)a+eu2(t)+r2(t)−r2(t)k2(t)eu1(t)]dt, | (2.7) |
ˉd2ω=∫ω0β(t−τ)eu2(t−τ)+u3(t−τ)−u3(t)a+eu2(t−τ)dt. | (2.8) |
From the Eqs (2.3) and (2.6), we have
∫ω0|˙u1(t)|dt=λ∫ω0|r1(t)(1−eu1(t)k1(t))+σ2(t)eu2(t)−u1(t)−σ1(t)|dt<∫ω0|r1(t)−r1(t)k1(t)eu1(t)+σ2(t)eu2(t)−u1(t)|dt+∫ω0|σ1(t)|dt<ˉσ1ω+ˉσ1ω=2ˉσ1ω, |
that is
∫ω0|˙u1(t)|dt<2ˉσ1ω. | (2.9) |
Similarly, it follows from (2.4) and (2.7), (2.5) and (2.8) that
∫ω0|˙u2(t)|dt<2ˉσ2ω, | (2.10) |
∫ω0|˙u3(t)|dt<2ˉd2ω, | (2.11) |
Since (u1(t),u2(t),u3(t))∈U, there exists ξi,ηi∈[0,ω], such that
ui(ξi)=mint∈[0,ω]ui(t),ui(ηi)=maxt∈[0,ω]ui(t),i=1,2,3. |
Multiplying (2.5) by eu3(t) and integrating over [0,ω], we have
∫ω0d2(t)eu3(t)dt=∫ω0β(t−τ)eu2(t−τ)+u3(t−τ)a+eu2(t−τ)dt. |
Now we make the change of a variable j=t−τ and Lemma2, we obtain
∫ω0β(t−τ)eu2(t−τ)+u3(t−τ)a+eu2(t−τ)dt=∫ω−τ−τβ(j)eu2(j)+u3(j)a+eu2(j)dj=∫ω0β(t)eu2(t)+u3(t)a+eu2(t)dt, |
that is
∫ω0d2(t)eu3(t)dt=∫ω0β(t)eu2(t)+u3(t)a+eu2(t)dt. | (2.12) |
From the Eq (2.12), we obtain
dL2∫ω0eu3(t)dt≤∫ω0d2(t)eu3(t)dt=∫ω0β(t)eu2(t)+u3(t)a+eu2(t)dt≤βMeu2(η2)a+eu2(ξ2)∫ω0eu3(t)dt, |
which is
dL2≤βMeu2(η2)a+eu2(ξ2), |
thus
u2(η2)≥lndL2(a+eu2(ξ2))βM. | (2.13) |
It follows from (2.10), (2.13) and Lemma3 that we have
u2(t)≥u2(η2)−∫ω0|˙u2(t)|dt>lndL2(a+eu2(ξ2))βM−2ˉσ2ω:=B21. | (2.14) |
In particular, we obtain
u2(ξ2)>lndL2(a+eu2(ξ2))βM−2ˉσ2ω, |
or
(βMe2ˉσ2ω−dL2)eu2(ξ2)−adL2>0. |
And in view of (H1), we have
u2(ξ2)>lnadL2βMe2ˉσ2ω−dL2=lnl−. |
In a similar way, from the Eq (2.12) we obtain
dM2≥βLeu2(ξ2)a+eu2(η2), |
therefore
u2(ξ2)≤lndM2(a+eu2(η2))βL. | (2.15) |
It follows from (2.10), (2.15) and Lemma3 that we have
u2(t)≤u2(ξ2)+∫ω0|˙u2(t)|dt<lndM2(a+eu2(η2))βL+2ˉσ2ω:=B22. | (2.16) |
In particular, we have
u2(η2)<lndM2(a+eu2(η2))βL+2ˉσ2ω, |
or
(βLe−2ˉσ2ω−dM2)eu2(η2)−adM2<0. |
In view of (H1), we have
u2(η2)<lnadM2βLe−2ˉσ2ω−dM2=lnl+. |
In view of (H2), we have
adM2βLe−2ˉσ2ω−dM2≥adL2βMe−2ˉσ2ω−dL2. |
It follows from (2.14) and (2.16) that
maxt∈[0,ω]|u2(t)|=maxt∈[0,ω]{|B21|,|B22|}:=B2. |
It follows from (2.6) that
ˉσ1ω≤ˉσ2ωelnl+eu1(ξ1)+ˉr1ω−¯(r1k1)ωeu1(ξ1), |
thus
u1(ξ1)≤ln{12(¯k1r1)[(ˉr1−ˉσ1)+√(ˉr1−ˉσ1)2+4ˉσ2l+(¯r1k1)]}=lnb1. |
This combined with (2.9), give
u1(t)≤u1(ξ1)+∫ω0|˙u1(t)|dt<lnb1+2ˉσ1ω:=B11. | (2.17) |
Similarly, we have
ˉσ1ω≥ˉr1ω−¯(r1k1)ωeu1(η1), |
therefore
u1(η1)≥ln[(¯k1r1)(ˉr1−ˉσ1)]=lnb2. |
This together with (2.9), gives
u1(t)≥u1(η1)−∫ω0|˙u1(t)|dt>lnb2−2ˉσ1ω:=B12. | (2.18) |
It follows from (2.17) and (2.18) that
maxt∈[0,ω]|u1(t)|=maxt∈[0,ω]{|B11|,|B12|}:=B1. |
It follows from that (2.7) that
ˉσ2ω≥−a−1ˉαωeu3(η3)+ˉr2ω−¯(r2k2)ωl+, |
thus
u3(η3)≥ln{aˉα−1[(ˉr2−ˉσ2)−(¯r2k2)l+]}=lnb3. |
This combined with (2.11), give
u3(t)≥u3(η3)−∫ω0|˙u3(t)|dt>lnb3−2ˉd2ω:=B31, | (2.19) |
Similarly,
ˉσ2ω≤−ˉαωeu3(ξ3)a+l++ˉσ1ωl−eu1(η1)+ˉr2ω, |
therefore
eu3(ξ3)≤(a+l+)[(ˉr2−ˉσ2)+ˉσ1l−eu1(η1)], |
or
u3(ξ3)<ln{ˉα−1(a+l+)[(ˉr2−ˉσ2)+ˉσ2l−eB1]}=lnb4. |
Combining with (2.11), give
u3(t)<u3(ξ3)+∫ω0|˙u3(t)|dt<lnb4+2ˉd2ω:=B32. | (2.20) |
It follows from (2.19) and (2.20) that
maxt∈[0,ω]|u3(t)|=maxt∈[0,ω]{|B31|,|B32|}:=B3. |
Next, let's consider QNu with u=(u1,u2,u3)∈R3. Note that
QN(u1,u2,u3)=[(ˉr1−ˉσ1)+ˉσ2eu2(t)−u1(t)−¯(r1k1)eu1(t), |
(ˉr2−ˉσ2)+ˉσ1eu1(t)−u2(t)−¯(r2k2)eu2(t)−ˉαeu3(t)a+eu2(t),−ˉd2+ˉβeu2(t)a+eu2(t)]. |
In view of (H1),(H2),(H3), QN(u1,u2,u3)=0 has a solution ˜u=(lnb5,lnu0,lnb6). Take B=max{B1+C,B2+C,B3+C}, where C>0 is taken sufficiently large such that ‖(lnb5,lnu0,lnb6)‖<C. Define Ω={u(t)=(u1(t),u2(t),u3(t))T∈U:‖u‖<B}. Then Ω is a bounded open subset of U, therefore Ω satisfies the requirement (a) in Lemma1. Moreover, it's not difficult to verify QNu≠0 for u∈∂Ω⋂kerL=∂Ω⋂R3. A direct computation gives deg{JQN,Ω⋂kerL,0}≠0. Therefore, system (2.1) has at least one ω periodic solution ˜u.
Step 2: We prove that the third equation of system (1.2) has a unique ω-periodic solution associated with the obtained ˜u. Letting
h(t)=β(t)y(t)z2(t)a+y(t)−β(t−τ)y(t−τ)z2(t−τ)a+y(t−τ), |
then the third equation of (1.2) is
dz1dt=−d1(t)z1(t)+h(t). | (2.21) |
Obviously,
d1(t+ω)=d1(t) |
and
h(t+ω)=β(t+ω)y(t+ω)z2(t+ω)a+y(t+ω)−β(t+ω−τ)y(t+ω−τ)z2(t+ω−τ)a+y(t+ω−τ)=β(t)y(t)z2(t)a+y(t)−β(t−τ)y(t−τ)z2(t−τ)a+y(t−τ)=h(t). |
Since d1(t) is nonnegative, ˉd1>0, it follows that
dz1dt=−d1(t)z1(t), | (2.22) |
admits exponential dichotomy. Therefore, we have
z1(t)=∫t−∞e−∫tsd1(σ)dσh(s)ds. |
Consequently, (lnx(t),lny(t),z1(t),lnz2(t)) is a ω-periodic solution of system (1.2). This completes the proof.
As an example, corresponding to the model (1.2), we have the following stage-structured predator-prey model with Holling type Ⅱ functional response incorporating prey refuge with actual biological parameters:
{dxdt=(1.2+sin20πt)x(t)(1−x(t)10+sin20πt)−0.2x(t)+0.15y(t),dydt=(1.5+sin20πt)y(t)(1−y(t)15+sin20πt)+0.2x(t)−0.15y(t)−(2+sin20πt)y(t)z2(t)2+y(t),dz1dt=(1.5+sin20πt)y(t)z2(t)2+y(t)−(1.5+sin20π(t−1))y(t−1)z2(t−1)2+y(t−1)−0.2z1(t),dz2dt=(1.5+sin20π(t−1))y(t−1)z2(t−1)2+y(t−1)−0.1z2(t), | (3.1) |
where r1(t)=1.2+sin20πt and r2(t)=1.5+sin20πt denote intrinsic growth rates, they are directly proportional to the the density of the prey x and y; k1(t)=10+sin20πt and k2(t)=15+sin20πt denote the environment carrying capacity for the prey x and y; σ1(t)=0.2 and σ2(t)=0.15 denote the per unit migration of the prey in the refuge to the predatory region and the opposite of it; α(t)=2+sin20πt denotes the maximal predator per capita consumption rate and β(t)=1.5+sin20πt denotes the rate of the predator consumes prey; (2+sin20πt)y(t)2+y(t) denotes Holling type Ⅱ functional response, which reflects the capture ability of the predator; d1(t)=0.2 and d2(t)=0.1 are the death rate of predator at any time t; (1.5+sin20πt)y(t−1)z2(t−1)/(2+y(t−1)) stands for the number of immature predator born at time (t−1) which still survive at time t and become mature predator.
Since τ=1, a=2, d2=0.1, σ2=0.15, r2(t)=1.5+sin20πt, k(t)=15+sin20πt, α(t)=2+sin20πt, β(t)=1.5+sin20πt, we have ˉd2=dM2=dL2=0.1,ˉσ2=0.15,ˉr2=1.5,¯(r2k2)=0.1,ˉα=2,βL=0.5,βM=2.5,ω=0.1. Simple computation shows
βMe2ˉσ2ω=2.5×e2×0.15×0.1=2.5×e0.03>2.5>0.1=dL2, |
(dL2)−1dM2βM=10×0.1×2.5=2.5>0.5=βL |
and
ˉαa−1eB3+(¯r2k2)l++ˉσ2 |
=12×2×emax{|B31|,|B32|}+110×2×0.10.5×e−0.03−0.1+0.15 |
=eB32+250e0.03−10+0.15 |
>(1.35+(2.5e0.03−0.01)emax{|B11|,|B12|})e0.02+250e0.03−10+0.15 |
>(1.35+(2.5e0.03−0.01)e0)e0.02+250e0.03−10+0.15>1.5=ˉr2. |
The above inequalities show that system (3.1) satisfies the hypothesis (H1),(H2),(H3) in Theorem1. Therefore, system (3.1) has at least one positive periodic solution.
In mathematical biology, dynamic relationship between predator and prey is always and will continue to be one of the main themes, many researchers have contributed to the study and improvement for the predator-prey model [17,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. The model in the present paper mimics the dynamic nature of the refuge. The populations varies due to the rates of emergence from and re-entry into refuge, which incorporates simultaneous effects of the refuge and migration of the population from the refuge area to the predatory area. For instance, it may happen in the birds migration. In fact, the dynamic nature of the refuge is an effective strategy for reducing predation as a prey population evolved. For this reason, a delayed stage-structured predator-prey model with a prey refuge is considered in this paper.
This work was supported by the National Natural Science Foundation of China under Grant (No. 11931016), Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016). The authors thank the editor and anonymous referees for their valuable suggestions and comments, which improved the presentation of this paper.
The authors declare that there is no conflict of interests regarding the publication of this article.
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