 Research
 Open Access
 Published:
Existence of nonconstant positive stationary solutions of the shadow predatorprey systems with Allee effect
Advances in Difference Equations volume 2014, Article number: 226 (2014)
Abstract
In this paper, we consider the dynamics of the shadow system of a kind of homogeneous diffusive predatorprey system with a strong Allee effect in prey. We mainly use the timemapping methods to prove the existence and nonexistence of the nonconstant positive stationary solutions of the system in the one dimensional spatial domain. The problem is assumed to be subject to homogeneous Neumann boundary conditions.
MSC:35K57, 35B09.
1 Introduction
In this paper, we are mainly concerned with the following homogeneous diffusive predatorprey system with a strong Allee effect in prey:
Here $u=u(x,t)$ and $v=v(x,t)$ stand for the densities of the prey and predator at time $t>0$ and a spatial position $x\in (0,h\pi )$ with $h\in (0,\mathrm{\infty})$, respectively; ${d}_{1},{d}_{2}>0$ are the diffusion coefficients of the species; d is the death rate of the predator, a measures the saturation effect, m is the strength of the interaction. The Allee threshold b is assumed to be smaller than 1. The strong Allee effect introduces a population threshold, and the population must surpass this threshold to grow. The boundary condition here is assumed to be homogeneous Neumann type, which implies that there is no flux for the populations on the boundary. For more details on the problem (1.1), we refer interested readers to [1–7] and references therein.
In [8], the authors considered the traveling wave solutions of system (1.1). More precisely, they showed that there is a nonnegative traveling wave solution of system (1.1) connecting the semitrivial solution $(b,0)$ and the positive equilibrium solution $({u}_{\ast},{v}_{\ast})$. They also proved that, under certain suitable conditions, there is a small traveling wave train solution of system (1.1).
In [7], the authors considered the nonexistence of nonconstant positive steady state solutions, and bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied. These results allow for the phenomenon that the rich impact of the Allee effect essentially increases the system spatiotemporal complexity.
Although the existence and nonexistence of nonconstant steady state solutions of the system (1.1) has been considered in [7] for finite diffusion coefficients, no results have been reported to consider the existence and nonexistence of the positive nonconstant steady state solutions for the shadow system corresponding to the system (1.1). The shadow system we mentioned here stands for the system where one of the diffusion coefficients tends to infinity. The readers are referred to [9–11] for the earlier contributions on the shadow systems.
Thus, the purpose of this paper is to consider the existence and nonexistence of positive nonconstant solutions of the following elliptic equations:
where ${d}_{2}\to \mathrm{\infty}$.
The methods we used in the paper are standard timemapping methods (see [12] and references therein for precise details on timemapping methods). We hope that the results in the paper will allow for a clearer understanding of the rich dynamics of this particular pattern formation system. In Section 2, we state the derivation of the shadow system of the original reactiondiffusion system (1.1). In Section 3, we study the existence of the nonconstant stationary solutions of the shadow system; in Section 4, we end up our discussions by drawing some conclusions.
2 Derivation of the shadow system
Firstly, we state the following useful a priori estimate for the nonnegative solutions of system (1.1) obtained in [7]:
Lemma 2.1 Suppose that ${d}_{1},{d}_{2},a,b,d,m,h>0$, and that $(u(x),v(x))$ is a nonnegative steady state solution of (1.1). Then either $(u,v)$ is one of constant solutions: $(0,0)$, or $(b,0)$, or for $x\in [0,h\pi ]$, $(u(x),v(x))$ satisfies
For later use in our discussion, we rewrite the second equation of system (1.2) in the following way:
Lemma 2.1 implies that the nonnegative solutions of the system (1.2) is bounded. Then we know that ${v}_{xx}\to 0$ as ${d}_{2}\to \mathrm{\infty}$. Since our problem is of Neumann boundary condition type, v is a constant, say ρ. As ${d}_{2}\to \mathrm{\infty}$, there exists a positive number $C=C({d}_{1})>0$, such that
Thus,
Integrating the second equation from 0 to ℓπ, we obtain
Thus, system (1.2) reduces to the following single ρparameterized scalar reactiondiffusion equation:
subject to the additional condition (2.3) and the condition
3 Existence of nonconstant positive stationary solutions of the shadow system
In this section, we mainly concentrate on the existence of the nonconstant positive solutions of the reduced shadow system (2.4).
For the purpose of our investigations, we define
Then we introduce the following energy functional:
From (3.2), we can find that, for any $x\in (0,h\pi )$, ${E}^{\prime}(x)\equiv 0$, and $F(u(x))<F(\beta )=F(\tau )$, where $\beta :=u(0)$, and $\tau :=u(h\pi )$.
It follows that if $u=u(x)$ is a solution of (2.4), then $F(u)$ must attain its local minimal value at a point in $(\beta ,\tau )$.
We rewrite $f(u)$ as
where
We have the following lemma on the properties of the function $\ell (u)$ defined above.
Lemma 3.1 For any $u\in (0,b)\cup (1,\mathrm{\infty})$, we have $\ell (u)<0$, while $\ell (u)>0$ for any $u\in (b,1)$. In particular, $\ell (b)=\ell (1)=0$. Moreover,

1.
Suppose that $ab+ab<0$ holds. Then for any $u\in ({\lambda}_{\ast},{\lambda}^{\ast})$, $\ell (u)$ is increasing, while for any $u\in (0,{\lambda}_{\ast})\cup ({\lambda}^{\ast},\mathrm{\infty})$, $\ell (u)$ is decreasing, where
$$\begin{array}{r}{\lambda}^{\ast}:=\frac{b+1a+\sqrt{{(b+1a)}^{2}+3(ab+ab)}}{3},\\ {\lambda}_{\ast}:=\frac{b+1a\sqrt{{(b+1a)}^{2}+3(ab+ab)}}{3}.\end{array}$$(3.4) 
2.
Suppose that $ab+ab\ge 0$ holds. Then, for any $u\in (0,{\lambda}^{\ast})$, $\ell (u)$ is increasing, while for any $u\in ({\lambda}^{\ast},\mathrm{\infty})$, $\ell (u)$ is decreasing.
Proof It is obvious that for any $u\in (0,b)\cup (1,\mathrm{\infty})$, we have $\ell (u)<0$, while $\ell (u)>0$ for any $u\in (b,1)$. We can directly check that
Because ${\lambda}^{\ast}>{\lambda}_{\ast}>0$ provided that $ab+ab<0$, we conclude from (3.5) that $\ell (u)$ is increasing (${\ell}^{\prime}(u)>0$) for $u\in ({\lambda}_{\ast},{\lambda}^{\ast})$, while for any $u\in (0,{\lambda}_{\ast})\cup ({\lambda}^{\ast},\mathrm{\infty})$, $\ell (u)$ is decreasing (${\ell}^{\prime}(u)<0$). The second part of the lemma can be proved similarly. □
From Lemma 3.1, it follows that $\ell (u)$ attains its maximum value ${\ell}_{\ast}:=\ell ({\lambda}^{\ast})$ at $u={\lambda}^{\ast}$. If $\rho >{\ell}_{\ast}$ holds, then $f(u)<0$ for all $u\in (0,\mathrm{\infty})$. Thus, $F(u)$ does not has its minimal value point in $(0,\mathrm{\infty})$, which implies that the shadow system (2.4) does not possess positive nonconstant stationary solutions. Similarly, if $\rho ={\ell}_{\ast}$ holds, then system (2.4) does not also possess positive nonconstant stationary solutions.
Thus, in order for the shadow system to have nonconstant positive stationary solutions, we need to concentrate on the case when $\rho \in (0,{\ell}_{\ast})$.
In this case, there exist two zeros of $f(u)=0$, and we can denote them by
with $b<{u}_{}(\rho )<{\lambda}^{\ast}<{u}_{+}(\rho )<1$.
Since $f(u)<0$ for $0<u<{u}_{}(\rho )$, $f(u)>0$ for ${u}_{}(\rho )<u<{u}_{+}(\rho )$, it follows that $F(u)$ is convex in $(0,{u}_{+}(\rho ))$, and concave in $({u}_{}(\rho ),\mathrm{\infty})$, and $F(u)$ taking its local minimum value at $u={u}_{}(\rho )$. In other words, $F(u)$ is decreasing in $(0,{u}_{}(\rho ))\cup ({u}_{+}(\rho ),\mathrm{\infty})$, and increasing in $({u}_{}(\rho ),{u}_{+}(\rho ))$.
Thus, the problem admits solutions for some $h>0$ if and only if $\rho \in (0,{\ell}_{\ast})$ and we are now deriving the precise information on the suitable $h>0$ such that the problem has positive nonconstant stationary solutions.
If $F(0)\ge F({u}_{+}(\rho ))$ holds, then there exists a unique ${\beta}^{\ast}\in (0,{u}_{}(\rho ))$, such that $F({\beta}^{\ast})=F({u}_{+}(\rho ))$. Define
Then for any $\beta \in ({\beta}_{0},{u}_{}(\rho ))$, there exists a unique ${\beta}^{\ast \ast}\in ({u}_{}(\rho ),{u}_{+}(\rho ))$, such that $F({\beta}_{0})=F({\beta}^{\ast \ast})$.
By the definition of $E(x)={d}_{1}{({u}_{x}(x))}^{2}/2+F(u(x))$, and the fact that ${E}^{\prime}(x)\equiv 0$, we have
Fix $u(h\pi )=\tau $, and integrate (3.8); we have
In the following, we want to show the limit of $h(\beta )$ as $\beta \to {u}_{}(\rho )$. In fact, following the same argument in [12] (say, for example, pp.314316), one can verify that
Also following [12], for a given number $u\in (\beta ,\tau )$, we define $u=g(s)$ by the relation
Then $s={g}^{1}(u)$ is well defined and is strictly increasing in $(\beta ,\tau )$, since in this interval $F(u)$ is convex and takes a strict minimum at $u={u}_{}(\rho )$.
Let $p>0$ be given by
Then we have
where we make another change of variable, $s=pcost$, $0\le t\le \pi $.
By the same argument as on p.314 of [12], we have
where
By (3.1), we have
Then after direct calculations, we have
Clearly, the function ${f}^{\u2033}(u)=0$ has a unique root ${u}_{1}>0$ since ${f}^{\u2033}(0)>0$, ${f}^{\u2034}(0)<0$ and ${lim}_{u\to +\mathrm{\infty}}{f}^{\u2033}(u)=\mathrm{\infty}$.
On the other hand, by the properties of f, we know that f has two critical points in $(0,{u}_{}(\rho ))$, denoted by ${c}_{1}$ and ${c}_{2}$, that is to say, ${f}^{\prime}({c}_{1})={f}^{\prime}({c}_{2})=0$. Since ${f}^{\prime}(0)<0$, ${lim}_{x\to \pm \mathrm{\infty}}{f}^{\prime}(u)=\mathrm{\infty}$, and ${f}^{\prime}(u)$ has a unique positive critical point ${u}_{1}>0$, it follows that ${c}_{1}<{u}_{1}<{u}_{2}$.
Thus, the results in Lemma 4.4 in [12] hold true in our problem. That is, for any $u\in (0,{u}_{}(\rho ))$, we have $H(u)<0$, which together with (3.14) implies that ${g}^{\u2034}(s)>0$.
By (3.12), we have $dp/d\beta <0$. By (3.13), we have
Since
and ${h}^{\u2033}(p)>0$ due to ${g}^{\u2034}(s)>0$, it follows that ${h}^{\prime}(p)>0$ or equivalently $dh/dp>0$. This together with the fact that $dp/d\beta <0$, we can conclude that $dh/d\beta <0$.
Summarizing the analysis above, we can conclude the following.
Theorem 3.2 Let $\ell (u)$, ${\lambda}^{\ast}$ be defined in (3.3) and (3.4), and ${u}_{}(\rho )$ be defined in (3.6). Then the shadow system (2.4) with the condition (2.3) has no nonconstant positive stationary solutions if $h\ge {\ell}_{\ast}:=\ell ({\lambda}^{\ast})$, and has nonconstant positive stationary solutions if and only if $\rho \in (0,min\{{\ell}_{\ast},{C}_{1}\})$ and
4 Conclusions
In this paper, we studied the existence and nonexistence of the positive nonconstant stationary solutions of a shadow system corresponding to a kind of diffusive homogeneous predatorprey system with Holling typeII functional response and strong Allee effect in prey. We hope that the results in the paper will allow for the clearer understanding of the rich dynamics of this particular pattern formation system. Future work might include considering the qualitative behavior of the parabolic shadow system.
References
 1.
Boukal DS, Sabelis MW, Berec L: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapse. Theor. Popul. Biol. 2007, 72: 136147. 10.1016/j.tpb.2006.12.003
 2.
Holling CS: The components of predation as revealed by a study of small mammal predation of the Euorpean pine swalfy. Can. Entomol. 1959, 91: 293320. 10.4039/Ent912935
 3.
Lewis M, Karevia P: Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 1993, 43: 141158. 10.1006/tpbi.1993.1007
 4.
Morozov A, Petrovovskii S, Li B: Spatiotemporal complexity of patchy invasion in a predatorprey system with the Allee effect. J. Theor. Biol. 2006, 238: 1835. 10.1016/j.jtbi.2005.05.021
 5.
Nisbet RM, Gurney WSC: Modelling Fluctuating Populations. Wiley, New York; 1982.
 6.
Petrovovskii S, Morozov A, Li B: Regimes of biological invasion in a predatorprey system with the Allee effect. Bull. Math. Biol. 2005, 67: 637661.
 7.
Wang J, Shi J, Wei J: Dynamics and pattern formation in a diffusive predatoryprey system with strong Allee effect in prey. J. Differ. Equ. 2011, 251: 12761304. 10.1016/j.jde.2011.03.004
 8.
Hsu C, Yang C, Yang T, Yang T: Existence of traveling wave solutions for diffusive predatorprey type systems. J. Differ. Equ. 2012, 252: 30403075. 10.1016/j.jde.2011.11.008
 9.
Nishiura Y: Global structure of bifurcating solutions of some reactiondiffusion systems. SIAM J. Math. Anal. 1982, 13: 555593. 10.1137/0513037
 10.
Opial Z:Sur les périodes des solutions de I’équation différentielle ${x}^{\u2033}+g(x)=0$. Ann. Pol. Math. 1961, 10: 4972.
 11.
Keener J: Activators and inhibitors in pattern formation. Stud. Appl. Math. 1978, 59: 123.
 12.
Jang J, Ni W, Tang M: Global bifurcation and structure of Turing patterns in the 1D LengyelEpstein model. J. Dyn. Differ. Equ. 2004, 16: 297320. 10.1007/s108840042782x
Acknowledgements
The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions, which led to an improved presentation of the manuscript.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have equal contributions. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bao, Z., Liu, H. Existence of nonconstant positive stationary solutions of the shadow predatorprey systems with Allee effect. Adv Differ Equ 2014, 226 (2014). https://doi.org/10.1186/168718472014226
Received:
Accepted:
Published:
Keywords
 homogeneous diffusive predatorprey system
 strong Allee effect in prey
 shadow system
 nonconstant positive stationary solutions