# Multiple Periodic Solutions of a Nonautonomous Plant-Hare Model

Authors: Yongfei Gao; P. J. Y. Wong; Y. H. Xia; Xiaoqing Yuan
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the
existence of at leasttwo positive periodic solutions for a plant-hare model with toxin-determined
functional response (nonmonotone). Some new technique is used in this paper, because standard
arguments in the literature are not applicable.


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## Body

## 1. Introduction

In the past few decades, the classical predator-prey model has been well studied. Such classical predator-prey model has, however, been questioned by several biologists (e.g., see [1, 2]). Based on experimental data, Holling [3] has proposed several types of monotone functional responses g(x)=c(t)x,c(t)x/(m+x),c(t)x2/(m+x2),c(t)x/(m+ax+x2) for these and other models. However, this will not be appropriate if we explore the impact of plant toxicity on the dynamics of plant-hare interactions [4]. Recently, Gao and Xia [5] considered a nonautonomous plant-hare dynamical system with a toxin-determined functional response given by
(1)N˙(t)=r(t)N(t)[1-N(t)K]-C(N(t))P(t),P˙(t)=B(t)C(N(t))P(t)-d(t)P(t),
where
(2)C(N(t))=f(N(t))[1-f(N(t))4G],f(N(t))=eδN(t)1+heδN(t).
Here, N(t) denotes the density of plant at time t, P(t) denotes the herbivore biomass at time t, r(t) is the plant intrinsic growth rate at time t, d(t) is the per capita rate of herbivore death unrelated to plant toxicity at time t, B(t) is the conversion rate at time t, e is the encounter rate per unit plant, δ is the fraction of food items encountered that the herbivore ingests, K is the carrying capacity of plant, G measures the toxicity level, and h is the time for handing one unit of plant. The functions r(t), d(t), and B(t) are continuous, positive, and periodic with period ω, and e, δ, K, G, and h are positive real constants. For any continuous ω-periodic function F, we let
(3)F¯=1ω∫0ω‍F(t)dt.The topological degree of a mapping has long been known to be a useful tool for establishing the existence of fixed points of nonlinear mappings. In particular, a powerful tool to study the existence of periodic solution of nonlinear differential equations is the coincidence degree theory (see [6]). Many papers study the existence of periodic solutions of biological systems by employing the topological degree theory; see, for example, [7–12] and references cited therein. However, most of them investigated the classical predator-prey model or the models with Holling functional responses; see [7–10]. There is no paper studying the functional responses in model (1) except for [5]. Gao and Xia [5] have obtained some sufficient conditions for the existence of at least one positive periodic solution for the system (1). Unlike the traditional Holling Type II functional response, systems with nonmonotone functional responses are capable of supporting multiple interior equilibria and bistable attractors. Thus, for nonautonomous system (1), it is possible to find two periodic solutions of (1). However, to date there is no work done on the existence of multiple periodic solutions of (1). Therefore, in this paper we will establish the existence of at least two positive periodic solutions of (1). We will be using the continuation theorem of Mawhin's coincidence degree theory; to this end some novel estimation technique will be employed to obtain a priori bounds of unknown solutions to some operator equation, as the standard estimation techniques used in the literature are not applicable to the system (1) due to the term C(N(t)). We will elaborate this in Remark 3.


## 2. Existence of Multiple Positive Periodic Solutions

In this section, we will establish sufficient conditions for the existence of at least two positive periodic solutions of (1). We will first summarize in the following a few concepts and results from [6] that will be required later.LetX,Y be normed vector spaces, L:DomL⊂X→Y a linear mapping, and N:X→Y a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dimKerL=codimImL<+∞ and ImL is closed in Y. If L is a Fredholm mapping of index zero, there exist continuous projectors P:X→X and Q:Y→Y such that ImP=KerL and KerQ=ImL=Im(I-Q). It follows that L∣domL∩KerP:(I-P)X→ImL is invertible. We denote the inverse of that map by Kp. If Ω is an open bounded subset of X, then the mapping N will be called L-compact on  Ω¯ if QN(Ω¯) is bounded and Kp(I-Q)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphismJ:ImQ→KerL.Lemma 1 (see [6]).
LetΩ⊂X be an open bounded set. Let L be a Fredholm mapping of index zero and N  L-compact on Ω¯. Assume(a)
for eachλ∈(0,1), x∈∂Ω∩DomL, Lx≠λNx;(b)
for eachx∈∂Ω∩KerL, QNx≠0;(c)
deg{JQN,Ω∩KerL,0}≠0.
ThenLx=Nx has at least one solution in Ω¯∩DomL.To proceed, we note that (1) is equivalent to
(4)N˙(t)=N(t)[4GeδP(t)+(4Gh-1)e2δ2N(t)P(t)4G(1+heδN(t))2r(t)(1-N(t)K)mmmlm-4GeδP(t)+(4Gh-1)e2δ2N(t)P(t)4G(1+heδN(t))2],P˙(t)=P(t)[4GeδB(t)N(t)+(4Gh-1)e2δ2B(t)N2(t)4G(1+heδN(t))2mmmml-d(t)4GeδB(t)N(t)+(4Gh-1)e2δ2B(t)N2(t)4G(1+heδN(t))2].Throughout, we assume the following:(A1)
1/4h<G<1/3h;(A2)
4hd¯exp(2r¯ω)<B¯<4Gd¯h2/(4Gh-1).We further introduce six positive numbers which will be used later as follows:(5)h±=(eδB¯exp(-2r¯ω)-2heδd¯)±Δ12d¯h2e2δ2,l±=[4Gh2eδB¯exp(2r¯ω)-2heδ(4Gh2d¯-(4Gh-1)B¯)]±Δ22h2e2δ2[4Gh2d¯-(4Gh-1)B¯],u±=(4GeδB¯-8Gheδd¯)±Δ32[4Gd¯h2e2δ2-(4Gh-1)e2δ2B¯],
where
(6)Δ1=[eδB¯exp(-2r¯ω)-2heδd¯]2-4d¯2h2e2δ2,Δ2=[4Gh2eδB¯exp(2r¯ω)-2heδ(4Gh2d¯-(4Gh-1)B¯)]2-4h2e2δ2[4Gh2d¯-(4Gh-1)B¯]2,Δ3=(4GeδB¯-8Gheδd¯)2-16Gd¯[4Gd¯h2e2δ2-(4Gh-1)e2δ2B¯].
Under assumptions (A1) and (A2), it is not difficult to show that
(7)l-<u-<h-<h+<u+<l+.Theorem 2.
In addition to (A1) and (A2), suppose that(A3)
1-(1/K)exp(lnl++2r¯ω)>0.
Then system (4) has at least two positive ω-periodic solutions.Proof.
Since we are concerned with positive solutions of system (4), we make use of the change of variables
(8)N(t)=exp(u1(t)),P(t)=exp(u2(t)).
Then, system (4) can be rewritten as
(9)u˙1(t)=r(t)-r(t)Kexp(u1(t))-4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2,u˙2(t)=-d(t)+4GeδB(t)exp(u1(t))+(4Gh-1)e2δ2B(t)exp(2u1(t))4G(1+heδexp(u1(t)))2.
Take(10)X=Y={x=(u1,u2)T∈C(ℝ,ℝ2)∣x(t+ω)=x(t)}
and define
(11)∥x∥=maxt∈[0,ω]|u1(t)|+maxt∈[0,ω]|u2(t)|,x=(u1,u2)T∈XorY.
Here |·| denotes the Euclidean norm. Then X and Y are Banach spaces with the norm ∥·∥. For any x=(u1,u2)T∈X, by means of the periodicity assumption, we can easily check that
(12)r(t)-r(t)Kexp(u1(t))-4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2:=f1(t)∈C(ℝ,ℝ),-d(t)+4GeδB(t)exp(u1(t))+(4Gh-1)e2δ2B(t)exp(2u1(t))4G(1+heδexp(u1(t)))2:=f2(t)∈C(ℝ,ℝ)
are ω-periodic.
Set(13)L:DomL∩X,L(u1(t),u2(t))T=(du1(t)dt,du2(t)dt)T,
where DomL={(u1(t),u2(t))T∈C1(ℝ,ℝ2)}. Further, N:X→X is defined by
(14)N(u1u2)=(f1(t)f2(t)).
Define
(15)P(u1u2)=(u1u2)=(1ω∫0ω‍u1(t)dt1ω∫0ω‍u2(t)dt),(u1u2)∈X=Y.
It is not difficult to show that
(16)KerL={x∣x∈X,x=C0,C0∈ℝ2},ImL={y∣y∈Y,∫0ω‍y(t)dt=0}isclosedinY,dimKerL=codimImL=2,
and P and Q are continuous projectors such that
(17)ImP=KerL,KerQ=ImL=Im(I-Q).
It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:ImL→DomL∩KerP exists and is given by
(18)Kp(y)=∫0t‍y(s)ds-1ω∫0ω‍∫0t‍y(s)dsdt.
Then QN:X→Y and Kp(I-Q)N:X→X are, respectively, defined by
(19)QNx=(1ω∫0ω‍f1(t)dt,1ω∫0ω‍f2(t)dt)T,Kp(I-Q)Nx=∫0t‍Nx(s)ds-1ω∫0ω‍∫0t‍Nx(s)dsdt-(tω-12)∫0ω‍Nx(s)ds.
Clearly, QN and Kp(I-Q)N are continuous. By using the Arzelà-Ascoli Theorem, it is not difficult to prove that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X. Moreover, QN(Ω¯) is bounded. Therefore, N is L-compact on Ω¯ for any open bounded set Ω⊂X.
Now, we will search for two appropriate open bounded subsets in order to apply the continuation theorem.
Corresponding to the operator equationLx=λNx, λ∈(0,1), we have
(20)u˙1(t)=λ[4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2r(t)-r(t)Kexp(u1(t))-4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2],(21)u˙2(t)=λ[4GeδB(t)exp(u1(t))+(4Gh-1)e2δ2B(t)exp(2u1(t))4G(1+heδexp(u1(t)))2-d(t)+4GeδB(t)exp(u1(t))+(4Gh-1)e2δ2B(t)exp(2u1(t))4G(1+heδexp(u1(t)))2].
Suppose x=(u1(t),u2(t))T∈X is a solution of (20) and (21) for a certain λ∈(0,1). Integrating (20), (21) over the interval [0,ω], we obtain
(22)∫0ω‍r(t)Kexp(u1(t))dt+∫0ω‍4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2dt=r¯ω,(23)∫0ω‍4GeδB(t)exp(u1(t))+(4Gh-1)e2δ2B(t)exp(2u1(t))4G(1+heδexp(u1(t)))2dt=d¯ω.
It follows from (A1), (20), and (22) that
(24)∫0ω‍|u˙1(t)|dt=λ∫0ω‍|[4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2r(t)-r(t)Kexp(u1(t))-4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2]|dt<∫0ω‍r(t)dt+∫0ω‍r(t)Kexp(u1(t))dt+∫0ω‍4Geδexp(u2(t))+(4Gh-1)e2δ2exp(u1(t)+u2(t))4G(1+heδexp(u1(t)))2dt=∫0ω‍r(t)dt+r¯ω=2r¯ω;
that is,
(25)∫0ω‍|u˙1(t)|dt<2r¯ω.
Similarly, it follows from (A1), (21), and (23) that
(26)∫0ω‍|u˙2(t)|dt<2d¯ω.
Since (u1(t),u2(t))T∈X, there exist ξi,ηi∈[0,ω] such that
(27)ui(ξi)=mint∈[0,ω]ui(t),ui(ηi)=maxt∈[0,ω]ui(t),i=1,2.
From (A1) and (23), we see that
(28)d¯ω≤∫0ω‍4GeδB(t)exp(u1(t))4G(1+heδexp(u1(t)))2dt+∫0ω‍(4Gh-1)e2δ2B(t)exp(2u1(t))4Gh2e2δ2exp(2u1(t))dt,
which implies
(29)d¯≤eδB¯exp(u1(η1))(1+heδexp(u1(ξ1)))2+(4Gh-1)B¯4Gh2.
So
(30)u1(η1)≥ln[4Gh2d¯-(4Gh-1)B¯](1+heδexp(u1(ξ1)))24Gh2eδB¯.
This, combined with (25), gives
(31)u1(t)≥u1(η1)-∫0ω‍|u˙1(t)|dt>ln[4Gh2d¯-(4Gh-1)B¯](1+heδexp(u1(ξ1)))24Gh2eδB¯-2r¯ω.
In particular, we have
(32)u1(ξ1)>ln[4Gh2d¯-(4Gh-1)B¯](1+heδexp(u1(ξ1)))24Gh2eδB¯-2r¯ω,
or
(33)[4Gh2d¯-(4Gh-1)B¯]h2e2δ2exp(2u1(ξ1))-[4Gh2eδB¯exp(2r¯ω)-2heδ(4Gh2d¯-(4Gh-1)B¯)]exp(u1(ξ1))+[4Gh2d¯-(4Gh-1)B¯]<0.
In view of (A2), we have
(34)lnl-<u1(ξ1)<lnl+.
Similarly, it follows from(A1) and (23) that
(35)d¯ω≥∫0ω‍4GeδB(t)exp(u1(t))4G(1+heδexp(u1(t)))2dt,
which implies
(36)d¯≥eδB¯exp(u1(ξ1))(1+heδexp(u1(η1)))2.
So
(37)u1(ξ1)≤lnd¯(1+heδexp(u1(η1)))2eδB¯.
This, combined with (25), gives
(38)u1(t)≤u1(ξ1)+∫0ω‍|u˙1(t)|dt<lnd¯(1+heδexp(u1(η1)))2eδB¯+2r¯ω.
In particular, we have
(39)u1(η1)<lnd¯(1+heδexp(u1(η1)))2eδB¯+2r¯ω,
or
(40)d¯h2e2δ2exp(2u1(η1))-(eδB¯exp(-2r¯ω)-2heδd¯)exp(u1(η1))+d¯>0.
It follows from (A2) that
(41)u1(η1)<lnh-oru1(η1)>lnh+.
From (25) and (34), we find
(42)u1(t)≤u1(ξ1)+∫0ω‍|u˙1(t)|dt<lnl++2r¯ω≜H11.
On the other hand, it follows from(A1), (22), and (42) that
(43)r¯ω≥∫0ω‍4Geδexp(u2(ξ2))4G(1+heδexp(lnl++2r¯ω))2dt,(44)r¯ω≤∫0ω‍r(t)Kexp(lnl++2r¯ω)dt+∫0ω‍eδexp(u2(η2))dt+∫0ω‍eδexp(u2(η2))2dt.
It follows from (43) that
(45)u2(ξ2)≤lnr¯(1+heδexp(lnl++2r¯ω))2eδ.
This, combined with (26), gives
(46)u2(t)≤u2(ξ2)+∫0ω‍|u˙2(t)|dt<lnr¯(1+heδexp(lnl++2r¯ω))2eδ+2d¯ω≜H21.
Moreover, because of (A3), it follows from (44) that
(47)u2(η2)≥ln2r¯(1-(1/K)exp(lnl++2r¯ω))3eδ.
This, combined with (26) again, gives
(48)u2(t)≥u2(η2)-∫0ω‍|u˙2(t)|dt>ln2r¯(1-(1/K)exp(lnl++2r¯ω))3eδ-2d¯ω≜H22.
It follows from (46) and (48) that
(49)maxt∈[0,ω]u2(t)<max{|H21|,|H22|}≜H2.
Now, let us considerQNx with x=(u1,u2)T∈ℝ2. Note that
(50)QN(u1,u2)T=(4GeδB¯exp(u1)+(4Gh-1)e2δ2B¯exp(2u1)4G(1+heδexp(u1))2r¯-r¯Kexp(u1)-4Geδexp(u2)+(4Gh-1)e2δ2exp(u1+u2)4G(1+heδexp(u1))2,-d¯+4GeδB¯exp(u1)+(4Gh-1)e2δ2B¯exp(2u1)4G(1+heδexp(u1))2)T.
Noting (A1), (A2), and (A3), we can show that the equation QN(u1,u2)T=0 has two distinct solutions:
(51)u~=(lnu-,ln4G(r¯-(r¯/K)u-)(1+heδu-)24Geδ+(4Gh-1)e2δ2u-),u^=(lnu+,ln4G(r¯-(r¯/K)u+)(1+heδu+)24Geδ+(4Gh-1)e2δ2u+).
Choose C>0 such that
(52)C>max{|ln4G(r¯-(r¯/K)u-)(1+heδu-)24Geδ+(4Gh-1)e2δ2u-|,mmlll|ln4G(r¯-(r¯/K)u+)(1+heδu+)24Geδ+(4Gh-1)e2δ2u+|}.
We are now ready to define two open bounded subsets in order to apply the continuation theorem. Let(53)Ω1={maxt∈[0,ω]x=(u1,u2)T∈X∣u1(t)∈(lnl-,lnh-),mmmlmaxt∈[0,ω]|u2(t)|<H2+C},Ω2={maxt∈[0,ω]x=(u1,u2)T∈X∣mint∈[0,ω]u1(t)∈(lnl-,lnl+),mmmlmaxt∈[0,ω]u1(t)∈(lnh+,H11),maxt∈[0,ω]|u2(t)|<H2+C}.
Then both Ω1 and Ω2 are bounded open subsets of X. It follows from (4) and (52) that u~∈Ω1 and u^∈Ω2. With the help of (4), (34), (41), (42), (49), and (52), it is easy to see that Ω1∩Ω2=ϕ and Ωi satisfies the requirement (a) in Lemma 1 for i=1,2. Moreover, QNx≠0 for x∈∂Ω∩KerL=∂Ω∩ℝ2. A direct computation gives deg{JQN,Ωi∩KerL,0}≠0. Here, J is taken as the identity mapping since ImQ=KerL. So far we have proved that Ωi satisfies all the assumptions in Lemma 1. Hence, (4) has at least two  ω-periodic solutions. This completes the proof of Theorem 2.Remark 3.
In the proof of Theorem2, we have employed some new technique to obtain a priori bounds for u1. Here, the standard arguments in the literature (see, e.g., [7–12]) do not work. Indeed, from (23) in the proof it follows that
(54)d¯ω≤4GeδB¯ωexp(u1(η1))+4Ghe2δ2B¯ωexp(2u1(η1))4G(1+heδexp(u1(ξ1)))2.
If we were to use the standard arguments in the literature, then we have
(55)[4d¯h3e2δ2B¯-4h2e2δ2B¯2exp(4r¯ω)]exp(2u1(ξ1))+[8d¯h2eδB¯-4heδB¯2exp(2r¯ω)]exp(u1(ξ1))+4d¯hB¯<0,
where u1(ξ1)=mint∈[0,ω]u1(t) and u1(η1)=maxt∈[0,ω]u1(t). It follows from (55) that
(56)l~-<exp(u1(ξ1))<l~+,
where l~- and l~+ are the roots of the following equation in x:
(57)[4d¯h3e2δ2B¯-4h2e2δ2B¯2exp(4r¯ω)]x2+[8d¯h2eδB¯-4heδB¯2exp(2r¯ω)]x+4d¯hB¯=0.
We claim that (57) has at least a negative root; that is, at least one of l~-,  l~+ is negative. Otherwise, if both l~- and l~+ are positive, then from (57) we see that
(58)l~+·l~-=4d¯hB¯4d¯h3e2δ2B¯-4h2e2δ2B¯2exp(4r¯ω)>0,
which implies
(59)hd¯>B¯exp(4r¯ω).
On the other hand, it follows form (57) and (58) that
(60)l~++l~-=-8d¯h2eδB¯-4heδB¯2exp(2r¯ω)4d¯h3e2δ2B¯-4h2e2δ2B¯2exp(4r¯ω)<0,
which contradicts the positivity of l~- and l~+. Therefore, at least one of l~-,l~+ is negative. However, to use the standard arguments in the literature we need both l~- and l~+ to be positive. Hence, we have illustrated that standard arguments in the literature are not applicable to the system (4) and some new technique should be used. To see how this problem is handled, the reader may refer to (27)–(34) in the proof of Theorem 2.



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