
This paper aimed to study the influence of herd immigration in prey species on the stability of the prey–predator interaction, in which prey immigration is modeled as a herd movement for defensive purposes. A stochastic version of the model was formulated to incorporate the influence of random noises. Positivity and boundedness are discussed for both deterministic and stochastic models, which validate the model biologically. For the deterministic model, the local asymptotic stability of the feasible equilibrium points is discussed, and the Hopf bifurcation is exhibited with respect to an immigration factor. Using a suitable Lyapunov function, sufficient conditions for global asymptotic stability are established for deterministic and stochastic models. Numerical simulations are carried out to verify and clarify our analytical findings. It is demonstrated that increasing prey herd immigration rates stabilizes the systems. Numerical simulations of the stochastic system reveal that population density fluctuations grow more consistently as prey herd immigration increases; these simulations also exhibit diverse dynamics, including quasi-steady states and quasi-limit cycles. It is concluded that the immigration of prey herds improves the survival of both species in deterministic and stochastic systems. Thus, it may be beneficial for prey to immigrate in groups to support unstable systems.
Citation: Jawdat Alebraheem. Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration[J]. AIMS Mathematics, 2025, 10(3): 4620-4640. doi: 10.3934/math.2025214
[1] | Famei Zheng . Periodic wave solutions of a non-Newtonian filtration equation with an indefinite singularity. AIMS Mathematics, 2021, 6(2): 1209-1222. doi: 10.3934/math.2021074 |
[2] | Shanshan Yu, Jiang Liu, Xiaojie Lin . Multiple positive periodic solutions of a Gause-type predator-prey model with Allee effect and functional responses. AIMS Mathematics, 2020, 5(6): 6135-6148. doi: 10.3934/math.2020394 |
[3] | Weijie Lu, Yonghui Xia . Periodic solution of a stage-structured predator-prey model with Crowley-Martin type functional response. AIMS Mathematics, 2022, 7(5): 8162-8175. doi: 10.3934/math.2022454 |
[4] | Haitham Qawaqneh, Ali Altalbe, Ahmet Bekir, Kalim U. Tariq . Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential. AIMS Mathematics, 2024, 9(9): 23410-23433. doi: 10.3934/math.20241138 |
[5] | Tuersunjiang Keyoumu, Wanbiao Ma, Ke Guo . Existence of positive periodic solutions for a class of in-host MERS-CoV infection model with periodic coefficients. AIMS Mathematics, 2022, 7(2): 3083-3096. doi: 10.3934/math.2022171 |
[6] | Yun Xin, Hao Wang . Positive periodic solution for third-order singular neutral differential equation with time-dependent delay. AIMS Mathematics, 2020, 5(6): 7234-7251. doi: 10.3934/math.2020462 |
[7] | Abdullahi Yusuf, Tukur A. Sulaiman, Mustafa Inc, Sayed Abdel-Khalek, K. H. Mahmoud . M−truncated optical soliton and their characteristics to a nonlinear equation governing the certain instabilities of modulated wave trains. AIMS Mathematics, 2021, 6(9): 9207-9221. doi: 10.3934/math.2021535 |
[8] | Caifeng Liu . Linear Rayleigh-Taylor instability for compressible viscoelastic fluids. AIMS Mathematics, 2023, 8(7): 14894-14918. doi: 10.3934/math.2023761 |
[9] | Amna Mumtaz, Muhammad Shakeel, Abdul Manan, Marouan Kouki, Nehad Ali Shah . Bifurcation and chaos analysis of the Kadomtsev Petviashvili-modified equal width equation using a novel analytical method: describing ocean waves. AIMS Mathematics, 2025, 10(4): 9516-9538. doi: 10.3934/math.2025439 |
[10] | Yazid Alhojilan, Islam Samir . Investigating stochastic solutions for fourth order dispersive NLSE with quantic nonlinearity. AIMS Mathematics, 2023, 8(7): 15201-15213. doi: 10.3934/math.2023776 |
This paper aimed to study the influence of herd immigration in prey species on the stability of the prey–predator interaction, in which prey immigration is modeled as a herd movement for defensive purposes. A stochastic version of the model was formulated to incorporate the influence of random noises. Positivity and boundedness are discussed for both deterministic and stochastic models, which validate the model biologically. For the deterministic model, the local asymptotic stability of the feasible equilibrium points is discussed, and the Hopf bifurcation is exhibited with respect to an immigration factor. Using a suitable Lyapunov function, sufficient conditions for global asymptotic stability are established for deterministic and stochastic models. Numerical simulations are carried out to verify and clarify our analytical findings. It is demonstrated that increasing prey herd immigration rates stabilizes the systems. Numerical simulations of the stochastic system reveal that population density fluctuations grow more consistently as prey herd immigration increases; these simulations also exhibit diverse dynamics, including quasi-steady states and quasi-limit cycles. It is concluded that the immigration of prey herds improves the survival of both species in deterministic and stochastic systems. Thus, it may be beneficial for prey to immigrate in groups to support unstable systems.
During the past few decades, singular differential equations have been widely investigated by many scholars. Singular differential equations appear in many problems of applications such as the Kepler system describing the motion of planets around stars in celestial mechanics [12], nonlinear elasticity [10] and Brillouin focusing systems [2]. We refer to the classical monograph [31] for more information about the application of singular differential equations in science. Owing to the extensive applications in many branches of science and industry, singular differential equations have gradually become one of the most active research topics in the theory of ordinary differential equations. Up to this time, some necessary work has been done by scholars, including Torres [31,33], Mawhin [17], O'Regan [29], Ambrosetti [1], Fonda [12,13], Chu [5,8] and Zhang [36,37], etc.
In the current literature on singular differential equations, the problem on the existence and multiplicity of periodic solutions is one of the hot topics. Lazer and Solimini [25] first applied topological degree theory to study the periodic solutions of singular differential equations, the results also reveal that there are essential differences between repulsive singularity and attractive singularity. In order to avoid the collision between periodic orbit and singularity in the case of repulsive singularity, a strong force condition was first introduced by Gordon [16]. After that, various variational methods and topological methods based on topological degree theory have been widely used, including the method of upper and lower solutions [9,18,19,20], fixed point theorems [7], continuation theory of coincidence degree [21,24,26], and nonlinear Leray-schauder alternative principle [5,6,8]. From present literature, the existence periodic solutions are convenient to prove if the singular term satisfies the strong force condition. It is worth noting that Torres obtained existence results of periodic solutions in the case of a weak singularity condition of the singular term, see the reference [32] for details. Until now, the work on the existence of periodic solutions with weak conditions is much less than the work with strong force conditions, see [22,27,32].
Because of singular Rayleigh equations are widely applied in many fields, such as engineering technique, physics and mechanics fields [14,30]. Singular Rayleigh equations usually have multiple regulations and local periodic vibration phenomena. Hence, periodic solutions of singular Rayleigh equation becomes one key issue of singular Rayleigh equations. However, most of the results in the references [3,4,15,23,34,35] are concerned about one solution, while fewer works are concerned about multiple periodic solutions. Therefore, it is valuable to investigate the existence of multiple periodic solutions for singular Rayleigh equations in both theory and practice.
Motivated by the above literature, the main purpose of this paper is to verify the existence and multiplicity of periodic solutions of the following singular Rayleigh equation
x″+f(x′)+g(x)=e(t), | (1.1) |
where f∈C(R,R), e∈C(R/TZ,R), and g∈C((0,+∞),R) may be singular at the origin. We discuss both repulsive and attractive singularity with some weak conditions for the term g. It is said that Eq (1.1) has a repulsive singularity at the origin if
limx→0+g(x)=−∞ |
and has an attractive singularity at the origin if
limx→0+g(x)=+∞. |
The proof of the main results in this study is based on Mawhin's coincidence degree and the method of upper and lower solutions. Compared to the existing results about periodic problems of singular Rayleigh equations, the novelties lie in two aspects: (1) the singular term g has a weaker force condition; (2) the existence of arbitrarily many periodic solutions are concerned.
The rest of this paper is organized as follows. Some preliminary results are presented in Section 2. The main results will be presented and proved in Section 3. Finally, in Section 4, some examples and numerical solutions are expressed to illustrate the application of our results.
In this section, we first recall some basic results on the continuation theorem of coincidence degree theory [28].
Let X and Y be two real Banach spaces. A linear operator
L:Dom(L)⊂X→Y |
is called a Fredholm operator of index zero if
(i) ImL is a closed subset of Y,
(ii) dimKerL=codimImL<∞.
If L is a Fredholm operator of index zero, then there exist continuous projectors
P:X→X,Q:Y→Y |
such that
ImP=KerL,ImL=KerQ=Im(I−Q). |
It follows that
L|DomL∩KerP:(I−P)X→ImL |
is invertible and its inverse is denoted by KP.
If Ω is a bounded open subset of X, the continuous operator
N:Ω⊂X→Y |
is said to be L-compact in ˉΩ if
(iii) KP(I−Q)N(ˉΩ) is a relative compact set of X,
(iv) QN(ˉΩ) is a bounded set of Y.
Lemma 2.1. [28] Let Ω be an open and bounded set of X, L:D(L)⊂X→Y be a Fredholm operator of index zero and the continuous operator N:ˉΩ⊂X→Y be L-compact on ˉΩ. In addition, if the following conditions hold:
(A1) Lx≠λNx,∀(x,λ)∈∂Ω×(0,1),
(A2) QNx≠0,∀x∈KerL∩∂Ω,
(A3) deg{JQN,Ω∩KerL,0}≠0,
where J:ImQ→KerL is an homeomorphism map. Then Lx=Nx has at least one solution in ˉΩ.
In order to apply Lemma 2.1 to Eq (1.1), let X=C1T, Y=CT, where
C1T={x|x∈C1(R,R),x(t+T)=x(t)}, |
CT={x|x∈C(R,R),x(t+T)=x(t)}. |
Define
‖x‖=max{‖x‖∞,‖x′‖∞},‖x‖∞=maxt∈[0,T]|x(t)|. |
Clearly, X, Y are two Banach spaces with norms ‖⋅‖ and ‖⋅‖∞. Meanwhile, let
L:DomL={x:x∈C2(R,R)∩C1T}⊂X→Y,Lx=x″. |
Then
KerL=R,ImL={x:x∈Y,∫T0x(t)dt=0}, |
hence, L is a Fredholm operator of index zero. Define the projects P and Q by
P:X→X,[Px](t)=x(0)=x(T), |
Q:Y→Y,[Qx](t)=1T∫T0x(t)dt. |
Obviously,
ImP=KerL,KerQ=ImL. |
Let Lp=L|DomL⋂KerP, then Lp is invertible and its inverse is denoted by Kp:ImL→DomL,
[Kpx](t)=−tT∫T0(T−s)x(s)ds+∫t0(t−s)x(s)ds, |
Let N:X→Y, such that
[Nx](t)=−[f(x′(t))+g(x(t))]+e(t). |
It is easy to show that QN and KP(I−Q)N are continuous by the Lebesgue convergence theorem. By Arzela-Ascoli theorem, we get that QN(¯Ω) and Kp(I−Q)N(¯Ω) are compact for any open bounded set Ω in X. Therefore, N is L-compact on ¯Ω.
For the sake of convenience, we denote
mint∈[0,T]e(t)=e∗,maxt∈[0,T]e(t)=e∗,ω=T1q(‖e‖qc)1p−1. |
Moreover, we list the following condition
(H0) There exist two constants c>0 and p≥1, such that
f(x)⋅x≥c|x|p,∀(t,x)∈R2. |
Obviously, we have f(0)=0.
Lemma 3.1. Assume that x is a T-periodic solution of Eq (1.1). Then the following inequalities hold
g(x(s1))≥e(s1)≥e∗,g(x(t1))≤e(t1)≤e∗, |
where s1 and t1 be the maximum point and the minimum point of x(t) on [0,T].
Proof. Obviously,
x′(s1)=x′(t1)=0,x″(s1)≤0andx″(t1)≥0. |
Combining these with Eq (1.1), we get
g(x(s1))−e(s1)≥0,g(x(t1))−e(t1)≤0. |
Then we have
g(x(s1))≥e(s1)≥e∗,g(x(t1))≤e(t1)≤e∗. |
Theorem 3.2. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that
(H1) There exist only two positive constants ξ1 and η1 with η1>ξ1>ω, such that
g(ξ1)=e∗,g(η1)=e∗. |
Then Eq (1.1) has a positive T-periodic solution x1 satisfies
ξ1−ω≤x1(t)≤η1+ω,mint∈Rx1(t)≤η1,∀t∈R. | (3.1) |
Proof. Since Eq (1.1) can be written as an operator equation Lx=Nx, so we consider an auxiliary equation Lx=λNx,
x″(t)+λ[f(x′(t)+g(x(t))]=λe(t),λ∈(0,1). | (3.2) |
Suppose that x∈X is a periodic solution of the above equation. Multiplying both sides of Eq (3.2) by x′(t) and integrating on the interval [0,T], then we have
∫T0f(x′(t))x′(t)dt=∫T0e(t)x′(t)dt. |
By using the Hölder's inequality, it follows from (H0) and the above equality that
c‖x′‖pp≤∫T0f(x′(t))x′(t)dt=∫T0e(t)x′(t)dt≤‖e‖q‖x′‖p, |
where 1q+1p=1. Then we can obtain from the above inequality that
‖x′‖p≤(‖e‖qc)1p−1. | (3.3) |
By Lemma 3.1 and (H1), we can deduce that
x(s1)≥ξ1,x(t1)≤η1. | (3.4) |
Then, by (3.3) and (3.4), we have
|x(t)|=|x(t1)+∫tt1x′(s)ds|≤|x(t1)|+∫T0|x′(s)|ds≤|x(t1)|+(∫T0ds)1q⋅(∫T0|x′(s)|pds)1p≤|x(t1)|+T1q‖x′‖P≤η1+T1q(‖e‖qc)1p−1=η1+ω | (3.5) |
and
|x(t)|=|x(s1)+∫ts1x′(s)ds|≥|x(s1)|−∫T0|x′(s)|ds≥|x(s1)|−(∫T0ds)1q⋅(∫T0|x′(s)|pds)1p≥|x(s1)|−T1q‖x′‖P≥ξ1−T1q(‖e‖qc)1p−1=ξ1−ω. | (3.6) |
Combining with the above two inequalities, we get
ξ1−ω≤x(t)≤η1+ω,∀t∈[0,T]. | (3.7) |
By (H0) and the continuity of f, it is immediate to see that
f(x)≥0,ifx≥0andf(x)≤0,ifx<0. |
Therefore, let us define two sets
I1={t∈[0,T]|x′(t)≥0} |
and
I2={t∈[0,T]|x′(t)<0}. |
Integrating the Eq (3.2) over the sets I1, I2, we get
∫I1f(x′(t))dt+∫I1g(x(t))dt=∫I1e(t)dt |
and
∫I2f(x′(t))dt+∫I2g(x(t))dt=∫I2e(t)dt, |
which imply that
∫T0|f(x′(t))|dt≤∫T0|g(x(t))|dt+∫T0|e(t)|dt. | (3.8) |
Then by (3.2), (3.7) and (3.8), we obtain
|x′(t)|=|∫tt1x″(s)ds|≤∫T0|x″(t)|dt≤λ(∫T0|f(x′(t))|dt+∫T0|g(x(t))|dt+∫T0|e(t)|dt)<2(∫T0|g(x(t))|dt+∫T0|e(t)|dt)≤2T(gω+¯|e|):=M1, | (3.9) |
where
gω=maxξ1−ω≤x(t)≤η1+ω|g(x)| |
and ¯|e| is the mean value of |e(t)| on the interval [0,T].
Obviously, ξ1, η1 and M1 are positive constants independent of λ. Take three positive constants h1, h2 and ˜M1 with
h1<ξ1−ω<η1+ω<h2,˜M1>M1 | (3.10) |
and let
Ω1={x:x∈X,h1<x(t)<h2,|x′(t)|<˜M1,t∈[0,T]}. |
Obviously, Ω1 is an open bounded set of X. By the definition of N, we know that N is L-compact on the ˉΩ1. By (3.7), (3.9) and (3.10), we get that
x∈∂Ω1∩DomL,Lx≠λNx,λ∈(0,1). |
Hence, the condition (A1) in Lemma 2.1 is satisfied.
Next, we verify that the condition (A2) of Lemma 2.1 is satisfied. Clearly, if x∈∂Ω1∩KerL=∂Ω1∩R, we have QNx≠0. If it does not hold, then there exists x∈∂Ω1∩R, such that QNx=0, and x(t)≡ζ is a constant. That is
1T∫T0[−g(ζ)+e(t)]dt=0, |
i.e.,
g(ζ)−¯e=0. |
This implies that
e∗≤g(ζ)≤e∗, |
which together with (H1) yield
ζ∈[ξ1,η1]. |
This contradicts x=ζ∈∂Ω1∩R. Thus,
QNx≠0,∀x∈∂Ω1∩KerL. | (3.11) |
Finally, we prove that the condition (A3) of Lemma 2.1 is also satisfied. Define
H(μ,x)=μx+(1−μ)JQN(x), |
where
J=I:ImL→KerL,Jx=x. |
Then, by (3.11), we notice that
xH(μ,x)≠0,∀(μ,x)∈[0,1]×∂Ω1∩KerL. |
Therefore, we have
deg{JQNx,Ω1∩KerL,0}=deg{H(0,x),Ω1∩KerL,0}=deg{H(1,x),Ω1∩KerL,0}≠0. |
To sum up the above discussion, we have proven that all of the conditions of Lemma 2.1 are satisfied. Therefore, Eq (1.1) has a T-periodic solution x1 in Ω1. Moreover, by (3.4) and (3.7), we get that (3.1) holds.
Theorem 3.3. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that
(H2) There exist only four positive constants ξ1, ξ2, η1, η2 with ξ2>η2>η1>ξ1>ω, such that
g(ξ1)=e∗=g(ξ2),g(η1)=e∗=g(η2). |
Then Eq (1.1) has two positive T-periodic solutions x1 and x2, which satisfy
ξ1−ω≤x1(t)≤η1+ω,mint∈[0,T]x1(t)≤η1,∀t∈[0,T] | (3.12) |
and
η2≤x2(t)≤ξ2,∀t∈[0,T]. | (3.13) |
Proof. From Lemma 3.1 and (H2), we obtain
ξ1≤x(s1)≤ξ2 |
and
x(t1)≤η1orx(t1)≥η2. |
Therefore, we have either
η2≤x(t)≤ξ2,∀t∈[0,T] | (3.14) |
or
ξ1≤x(s1)≤ξ2,x(t1)≤η1. | (3.15) |
(1) If (3.14) holds, notice that
α(t)=η2 |
is a constant lower solution of Eq (1.1) and
β(t)=ξ2 |
is a constant upper solution of Eq (1.1). Then by the method of upper and lower solutions (see [11,31]), we know that Eq (1.1) has a positive T-periodic solution x2 such that (3.13) holds.
(2) If (3.15) holds, by (3.3), analysis similar to that in (3.5) and (3.6), we have
ξ1−ω≤x(t)≤η1+ω,∀t∈[0,T]. | (3.16) |
By using similar arguments of Theorem 3.2, it follows from (3.16) that there exists a constant M2>0 such that
|x′|<M2. |
Clearly, ξ1, η1, M2 are all independent of λ. Take three constants u1,u2, and ˜M2 with
0<u1<ξ1−ω<η1+ω<u2,˜M2>M2 |
and set
Ω2={x:x∈X,u1<x(t)<u2,|x′(t)|<˜M2,t∈[0,T],mint∈[0,T]x(t)≤η1}. |
The remainder can be proved in the same way as in the proof of Theorem 3.2. Then, Eq (1.1) has a positive T-periodic solution x1 in Ω2 such that (3.12) holds.
To sum up the above discussion, we plainly conclude that Eq (1.1) has at least two positive T-periodic solutions.
Remark 1. In Theorem 3.3, if η1=η2, we can only get that Eq (1.1) has at least one positive T-periodic solution.
Proof. If η1=η2, by Lemma 3.1 and (H2), we can only get
ξ1≤x(s1)≤ξ2. |
As in the proof of Theorem 3.3, we can prove that Eq (1.1) has a positive T-periodic solutions x1 such that
ξ1−ω≤x1(t)≤ξ2,maxt∈[0,T]x1(t)≥ξ1,forallt∈[0,T]. | (3.17) |
Moreover, by the method of upper and lower solutions (see [11,31]), we can also get that Eq (1.1) has a positive T-periodic solution x2 such that
η1=η2≤x2(t)≤ξ2,∀t∈[0,T]. | (3.18) |
But, by (3.17) and (3.18), we are not sure that x1 is different from x2.
Therefore, we just can assert that Eq (1.1) has at least one positive T-periodic solution.
Furthermore, Theorem 3.3 can be generalized to arbitrarily many periodic solutions.
Theorem 3.4. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that
(Hn) There exist only 2n positive constants ξ1, ξ2, ⋯ ξn, η1, η2, ⋯ ηn, with
ξn>ηn>ηn−1>ξn−1>⋯η1>ξ1>ω,ifniseven,ηn>ξn>ξn−1>ηn−1>⋯η1>ξ1>ω,ifnisodd | (3.19) |
and
η2i−1<ξ2i+1−ω,i=1,2⋯[n2], | (3.20) |
where [⋅] stands for the integer part, such that
g(ξ1)=g(ξ2)=⋯=g(ξn)=e∗, |
g(η1)=g(η2)=⋯=g(ηn)=e∗. |
Then Eq (1.1) has at least n different positive T-periodic solutions.
Proof. The case n=1 and n=2, one can see Theorem 3.2 and Theorem 3.3.
Let us define the following sets
B2i−1={x|x∈X,ξ2i−1−ω≤x(t)≤η2i−1+ω,maxt∈Rx(t)≥ξ2i−1mint∈Rx(t)≤η2i−1},i=1,2⋯[n+12],B2i={x|x∈X,η2i≤x(t)≤ξ2i,},i=1⋯[n2]. |
By (3.19) and (3.20), notice that Bi∩Bj=∅, for i≠j, i,j=1,2⋯n.
For the case n=3, by Lemma 3.1 and (H3), we have
ξ1≤x(s1)≤ξ2orx(s1)≥ξ3 |
and
x(t1)≤η1orη2≤x(t1)≤η3. |
Then
ξ1≤x(s1)≤ξ2andx(t1)≤η1 | (3.21) |
or
η2≤x(t)≤ξ2 | (3.22) |
or
x(s1)≥ξ3andη2≤x(t1)≤η3. | (3.23) |
By (3.21) and (3.22), as in the proof of Theorem 3.3, we can prove that Eq (1.1) has two different positive T-periodic solutions x1 and x2 with x1∈B1 and x2∈B2. By (3.23), analysis similar to that in the proof of Theorem 3.2 shows that Eq (1.1) has a positive T-periodic solution x3 belonging to B3. Then, by the facts, we get that Eq (1.1) has at least 3 different positive T-periodic solutions.
Similar arguments apply to the case n>3, we can prove that Eq (1.1) has n different positive T-periodic solutions x1, x2, ⋯ xn with xi∈Bi, i=1,2,⋯n.
The proof is completed.
Theorem 3.5. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that
(C1) There exist only two positive constants ξ1, η1 with η1>ξ1, such that
g(ξ1)=e∗,g(η1)=e∗. |
Then Eq (1.1) has at least one positive T-periodic solution.
Proof. Obviously,
α(t)=ξ1 |
is a constant lower solution of Eq (1.1) and
β(t)=η1 |
is a constant upper solution of Eq (1.1). Then by the method of upper and lower solutions (see [11,31]), we know that Eq (1.1) has a positive T-periodic solution x such that α(t)≤x(t)≤β(t) for every t.
Theorem 3.6. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that
(C2) There exist only four positive constants ξ1<η1<η2<ξ2, such that
g(ξ1)=e∗=g(ξ2),g(η1)=e∗=g(η2). |
Then Eq (1.1) has at least two positive T-periodic solutions.
Proof. The proof of Theorem 3.6 works almost exactly as the proof Theorem 3.3. It is easy to get that
x(s1)≤η1orx(s1)≥η2 |
and
ξ1≤x(t1)≤ξ2, |
which together with Lemma 3.1 yield that
ξ1≤x(t)≤η1,∀t∈[0,T] | (3.24) |
or
ξ1≤x(t1)≤ξ2,x(s1)≥η2. | (3.25) |
(1) If (3.24) holds, by the method of upper and lower solutions align (see [11,31]), we get that Eq (1.1) has at least one positive T-periodic solution x such that
ξ1≤x(t)≤η1,∀t∈[0,T]. |
(2) If (3.25) holds, repeating the proof of Theorem 3.2, we can construct an open bounded set
Ω3={x:x∈X,r1<x(t)<r2,|x′(t)|<˜M3,∀t∈[0,T]}, |
such that Eq (1.1) has at least one positive T-periodic solutions in Ω3.
To sum up the above discussion, we have proved that Eq (1.1) has at least two positive T-periodic solutions.
Similar as in the proof of Theorem 3.4, we can generalize Theorem 3.6 to arbitrarily many periodic solutions.
Theorem 3.7. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that
(Cn) There exist only 2n positive constants ξ1, ξ2, ⋯ ξn, η1, η2, ⋯ ηn with
ξn>ηn>ηn−1>ξn−1>⋯η1>ξ1>0,ifniseven,ηn>ξn>ξn−1>ηn−1>⋯η1>ξ1>0,ifnisodd |
and
ξ2i<η2i+2−ω,i=1,2⋯[n−22] |
such that
g(ξ1)=g(ξ2)=⋯=g(ξn)=e∗, |
g(η1)=g(η2)=⋯=g(ηn)=e∗. |
Then Eq (1.1) has at least n positive T-periodic solutions.
In this section, some examples and numerical solutions are given to illustrate the application of our results.
Example 1. Consider the following equation:
x″+13.2x′+3.3x−4x2=3.8sin(πt)+2. | (4.1) |
Conclusion: Eq (4.1) has at least one positive 2-periodic solution.
Proof. Corresponding to Eq (1.1), we have
f(x′)=13.2x′,g(x)=3.3x−4x2,e(t)=3.8sin(πt)+2. |
Obviously, e∗=5.8, e∗=−1.8. It is easy to see that there exist only two positive constants ξ1≈0.912, η1≈2.047 such that
g(ξ1)=e∗=−1.8,g(η1)=e∗=5.8. |
Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.2, we get that Eq (4.1) has at least one positive 2-periodic solution. Applying Matlab software, we obtain numerical periodic solution of Eq (4.1), which is shown in Figure 1.
Example 2. Consider the following equation:
x″+83x′−6.4x−1.7x2=−17.521+sin2(t). | (4.2) |
Conclusion: Eq (4.2) has at least two positive π-periodic solutions.
Proof. Corresponding to Eq (1.1), we have
f(x′)=83x′,g(x)=−6.4x−1.7x2,e(t)=−17.521+sin2(t). |
Obviously, e∗=−8.76, e∗=−17.52. It is easy to check that exist only four positive constants ξ1≈0.3323, η1≈0.5805, η2≈1.177, ξ2≈2.7 such that
g(ξ1)=g(ξ2)=e∗=−17.52,g(η1)=g(η2)=e∗=−8.76. |
Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.3, we get that Eq (4.2) has at least two positive π-periodic solutions. We obtain two numerical periodic solutions of Eq (4.2), which are shown in Figures 2 and 3, respectively.
Example 3. Consider the following equation:
x″+95(x′)3+5.6x−3x2=3sin(πt)+2. | (4.3) |
Conclusion: Eq (4.3) has at least one positive 2-periodic solution.
Proof. Corresponding to Eq (1.1), we have
f(x′)=95(x′)3,g(x)=5.6x−3x2,e(t)=3sin(πt)+2. |
Obviously, e∗=5, e∗=−1. It is easy to see that there exist only two positive constants ξ1≈0.757, η1≈1.24 such that
g(ξ1)=e∗=−1,g(η1)=e∗=5. |
Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.2, we get that Eq (4.3) has at least one positive 2-periodic solution. Applying Matlab software, we obtain numerical periodic solution of Eq (4.3), which is shown in Figure 4.
In this paper, we study the existence and multiplicity of positive periodic solutions of the singular Rayleigh differential equation (1.1). Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, we construct some subsets Bk, k=1,2,⋯,n of C1T with Bi∩Bj=∅, for i≠j, i,j=1,2,⋯,n, such that the equation (1.1) has a positive T-periodic solution in each set Bk, k=1,2,⋯,n. That is, the equation (1.1) has at least n distinct positive T-periodic solutions. We discuss both the repulsive singular case and the attractive singular case, and the singular term has a weaker force condition than the literatures about strong force condition. Some results in the literature are generalized and improved. It should be pointed out that it is the first time to study the existence of arbitrarily many periodic solutions of singular Rayleigh equations. In addition, some typical numerical examples and the corresponding simulations have been presented at the end of this paper to illustrate our theoretical analysis.
We would like to express our great thanks to the referees for their valuable suggestions. Zaitao Liang was supported by the Natural Science Foundation of Anhui Province (No. 1908085QA02), the National Natural Science Foundation of China (No.11901004) and the Key Program of Scientific Research Fund for Young Teachers of AUST (No. QN2018109). Hui Wei was supported by the Postdoctoral Science Foundation of Anhui Province (No. 2019B318) and the National Natural Science Foundation of China (No. 11601007).
The authors declare that they have no competing interests concerning the publication of this article.
[1] | A. J. Lotka, Elements of physical biology, Williams & Wilkins company, 1925. |
[2] | V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Memoria della Reale Accademia Nazionale dei Lincei, 2 (1926), 31–113. |
[3] |
A. A. Berryman, The orgins and evolution of predator‐prey theory, Ecology, 73 (1992), 1530–1535. https://doi.org/10.2307/1940005 doi: 10.2307/1940005
![]() |
[4] |
C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/Ent91293-5 doi: 10.4039/Ent91293-5
![]() |
[5] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/Ent91385-7 doi: 10.4039/Ent91385-7
![]() |
[6] |
C. S. Holling, The functional response of invertebrate predators to prey density, Memoirs Entomol. Soc. Can., 98 (1966), 5–86. https://doi.org/10.4039/entm9848fv doi: 10.4039/entm9848fv
![]() |
[7] |
J. Alebraheem, Y. A. Hasan, Dynamics of a two predator–one prey system, Comp. Appl. Math., 33 (2014), 767–780. https://doi.org/10.1007/s40314-013-0093-8 doi: 10.1007/s40314-013-0093-8
![]() |
[8] |
B. Xie, Impact of the fear and Allee effect on a Holling type Ⅱ prey–predator model, Adv. Differ. Equ., 2021 (2021), 464. https://doi.org/10.1186/s13662-021-03592-6 doi: 10.1186/s13662-021-03592-6
![]() |
[9] |
C. Arancibia-Ibarra, J. Flores, Dynamics of a Leslie–Gower predator–prey model with Holling type Ⅱ functional response, Allee effect and a generalist predator, Math. Comput. Simulat., 188 (2021), 1–22. https://doi.org/10.1016/j.matcom.2021.03.035 doi: 10.1016/j.matcom.2021.03.035
![]() |
[10] | L. L. Rockwood, Introduction to population ecology, Blackwell Pub, 2006. |
[11] |
J. Alebraheem, Paradox of enrichment in a stochastic predator-prey model, J. Math., 2020 (2020), 8864999. https://doi.org/10.1155/2020/8864999 doi: 10.1155/2020/8864999
![]() |
[12] |
B. Han, D. Jiang, Stationary distribution, extinction and density function of a stochastic prey-predator system with general anti-predator behavior and fear effect, Chaos Soliton. Fract., 162 (2022), 112458. https://doi.org/10.1016/j.chaos.2022.112458 doi: 10.1016/j.chaos.2022.112458
![]() |
[13] |
S. Zhang, S. Yuan, T. Zhang, A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments, Appl. Math. Comput., 413 (2022), 126598. https://doi.org/10.1016/j.amc.2021.126598 doi: 10.1016/j.amc.2021.126598
![]() |
[14] |
N. Al-Salti, F. Al-Musalhi, V. Gandhi, M. Al-Moqbali, I. Elmojtaba, Dynamical analysis of a prey-predator model incorporating a prey refuge with variable carrying capacity, Ecol. Complex., 45 (2021), 100888. https://doi.org/10.1016/j.ecocom.2020.100888 doi: 10.1016/j.ecocom.2020.100888
![]() |
[15] |
P. Panja, Combine effects of square root functional response and prey refuge on predator–prey dynamics, Int. J. Model. Simulat., 41 (2021), 426–433. https://doi.org/10.1080/02286203.2020.1772615 doi: 10.1080/02286203.2020.1772615
![]() |
[16] |
M. S. Rahman, M. S. Islam, S. Sarwardi, Effects of prey refuge with Holling type Ⅳ functional response dependent prey predator model, Int. J. Model. Simulat., 45 (2025), 20–38. https://doi.org/10.1080/02286203.2023.2178066 doi: 10.1080/02286203.2023.2178066
![]() |
[17] |
X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. https://doi.org/10.1007/s00285-016-0989-1 doi: 10.1007/s00285-016-0989-1
![]() |
[18] |
S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model, Math. Biosci. Eng., 16 (2019), 5146–5179. https://doi.org/10.3934/mbe.2019258 doi: 10.3934/mbe.2019258
![]() |
[19] |
Z. Zhu, R. Wu, L. Lai, X. Yu, The influence of fear effect to the Lotka-Volterra predator-prey system with predator has other food resource, Adv. Differ. Equ., 2020 (2020), 237. https://doi.org/10.1186/s13662-020-02612-1 doi: 10.1186/s13662-020-02612-1
![]() |
[20] |
Y. Shao, Global stability of a delayed predator–prey system with fear and Holling-type Ⅱ functional response in deterministic and stochastic environments, Math. Comput. Simulat., 200 (2022), 65–77. https://doi.org/10.1016/j.matcom.2022.04.013 doi: 10.1016/j.matcom.2022.04.013
![]() |
[21] |
X. Guan, Y. Liu, X. Xie, Stability analysis of a Lotka-Volterra type predator-prey system with Allee effect on the predator species, Commun. Math. Biol. Neurosci., 2018 (2018), 9. https://doi.org/10.28919/cmbn/3654 doi: 10.28919/cmbn/3654
![]() |
[22] |
D. Sen, S. Ghorai, S. Sharma, M. Banerjee, Allee effect in prey's growth reduces the dynamical complexity in prey-predator model with generalist predator, Appl. Math. Model., 91 (2021), 768–790. https://doi.org/10.1016/j.apm.2020.09.046 doi: 10.1016/j.apm.2020.09.046
![]() |
[23] |
U. Kumar, P. S. Mandal, Role of Allee effect on prey–predator model with component Allee effect for predator reproduction, Math. Comput. Simulat., 193 (2022), 623–665. https://doi.org/10.1016/j.matcom.2021.10.027 doi: 10.1016/j.matcom.2021.10.027
![]() |
[24] |
J. Alebraheem, Dynamics of a predator–prey model with the effect of oscillation of immigration of the prey, Diversity, 13 (2021), 23. https://doi.org/10.3390/d13010023 doi: 10.3390/d13010023
![]() |
[25] |
T. Tahara, M. K. A. Gavina, T. Kawano, J. M. Tubay, J. F. Rabajante, H. Ito, et al, Asymptotic stability of a modified Lotka-Volterra model with small immigrations, Sci. Rep., 8 (2018), 7029. https://doi.org/10.1038/s41598-018-25436-2 doi: 10.1038/s41598-018-25436-2
![]() |
[26] |
V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal.-Real, 12 (2011), 2319–2338. https://doi.org/10.1016/j.nonrwa.2011.02.002 doi: 10.1016/j.nonrwa.2011.02.002
![]() |
[27] |
P. A. Braza, Predator–prey dynamics with square root functional responses, Nonlinear Anal.-Real, 13 (2012), 1837–1843. https://doi.org/10.1016/j.nonrwa.2011.12.014 doi: 10.1016/j.nonrwa.2011.12.014
![]() |
[28] |
P. Panja, Combine effects of square root functional response and prey refuge on predator–prey dynamics, Int. J. Model. Simulat., 41 (2021), 426–433. https://doi.org/10.1080/02286203.2020.1772615 doi: 10.1080/02286203.2020.1772615
![]() |
[29] |
S. Debnath, P. Majumdar, S. Sarkar, U. Ghosh. Complex dynamical behaviour of a delayed prey-predator model with square root functional response in presence of fear in the prey, Int. J. Model. Simulat., 43 (2023), 612–637. https://doi.org/10.1080/02286203.2022.2107887 doi: 10.1080/02286203.2022.2107887
![]() |
[30] | X. Mao, Stochastic differential equations and applications, Chichester: Horwood, 1997. |
[31] |
N. Dalal, D. Greenhalgh, X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084–1101. https://doi.org/10.1016/j.jmaa.2007.11.005 doi: 10.1016/j.jmaa.2007.11.005
![]() |
[32] |
H. I. Freedman, G. S. k. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493–508. https://doi.org/10.1016/S0092-8240(86)90004-2 doi: 10.1016/S0092-8240(86)90004-2
![]() |
[33] |
J. Alebraheem, T. Q. Ibrahim, G. E. Arif, A. A. Hamdi, O. Bazighifan, A. H. Ali, The stabilizing effect of small prey immigration on competitive predator-prey dynamics, Math. Comp. Model. Dyn., 30 (2024), 605–625. https://doi.org/10.1080/13873954.2024.2366337 doi: 10.1080/13873954.2024.2366337
![]() |
[34] |
J. H. Brown, A. Kodric-Brown, Turnover rates in insular biogeography: Effect of immigration on extinction, Ecology, 58 (1977), 445–449. https://doi.org/10.2307/1935620 doi: 10.2307/1935620
![]() |
[35] | A. Eriksson, F. Elías-Wolff, B. Mehlig, A. Manica, The emergence of the rescue effect from explicit withinand between-patch dynamics in a metapopulation, Proc. Biol. Sci., 281 (1780) (2014), 20133127. https://doi.org/10.1098/rspb.2013.3127 |
[36] | O. Ovaskainen, The interplay between immigration and local population dynamics in metapopulations, Ann. Zool. Fennici., 54 (2017), 113–121. |
1. | Xing Hu, Yongkun Li, Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales, 2022, 6, 2504-3110, 121, 10.3390/fractalfract6020121 |