具有强Allee效应捕食—食饵系统的动力学性质分析
|
摘要
相互作用物种间的模式生成和分布在保护生态方面和生化反应方面有重要的意义,一种典型的作用就是捕食-食饵关系,或者更一般地说为消耗者-资源关系。而这种作用关系体现在数学上的常微分方程(反应扩散方程)组,泛函微分方程组也是非线性微分方程研究领域中重要的方向,其研究方法包括经典的动力系统理论,解析半群方法及拓扑方法等。
当种群具有强Allee效应增长时,此前只有数值上的模拟结果,本文主要对具有强Allee效应的捕食-食饵系统进行了系统的解析分析。不同于Logistic型增长的捕食-食饵系统,强Allee效应使系统具有双稳定性结构,因此很多常用的研究方法用起来存在困难或者不再适用,需要一些改进的方法及一定的构造性技巧,本文主要工作如下:
1.对常微分系统进行了详细的全局双稳定动力学分析,以正平衡点的第一个分量为参数,得到了两个全局分支值:异宿轨道环分支值和Hopf分支值。在参数不同的取值范围内分别得到了相应的全局稳定的零平衡点、唯一的异宿轨道环、全局双稳定的零平衡点和周期轨道、全局双稳定的零平衡点和正平衡点及全局双稳定的零平衡点和半平衡点。在证明周期解的不存在性时给出了改进的Dulac函数判断方法,这个结果在具体应用时更为适用。此外,本文还证明了具有其他类型Allee效应的一些捕食-食饵系统还有更丰富的性质,例如多个周期轨道。本文严格的分析可以应用到很多具体的强Allee效应的捕食-食饵模型。其解析结果几乎是最新,最完整的,且不依赖于非线性函数具体的代数形式,并对其它二维常微分方程组的动力学行为分析提供了系统的方法和途径。
2.通过构造上、下解,能量估计等方法研究了具强Allee效应的反应扩散方程组的基本动力学行为,得到了整体解的存在性及渐近性,给出了解的先验估计,并得到了系统的双稳定性及空间齐次和非齐次的周期解。特别地,当捕食者的初始值足够大时,解最终趋于(0,0),这也表明(0,0)总是局部稳定的,即给定食饵的初始值, (0,0)的吸引域包含所有充分大的.因此若存在其他的局部稳定的稳态解或者周期解,则系统是双稳定(或者多重稳定)的;当食饵的初始值小于强效应的门槛值时, (0,0)也是全局渐近稳定的,这是具有强Allee效应的捕食-食饵系统的特征。这些结果也表明强Allee效应从本质上增加了反应扩散方程组时空动力学行为的复杂性。
3.对具有强Allee效应的椭圆方程组进行了详细的分析。利用椭圆方程的正则性估计得到了非常数正稳态解的先验估计及不存在性;分析了平凡稳态解的稳定性和半平凡稳态解的分支情况;由于强Allee效应下的双稳定结构,且系统有非常多的半平凡稳态解分支,因而难以证得稳态解的正下界,常用的Leray-Schauder度理论不能够得到非常数的正稳态解,本文采用的是史峻平和王学峰[1]推广的全局分支理论得到非常数正稳态解的存在性,即空间的模式生成;
4.分别考虑带两个离散滞量的泛函和偏泛函微分方程组的稳定性和分支分析。讨论了滞量对强Allee效应捕食-食饵系统的不稳定性影响,及在此基础上扩散项对系统稳定性的影响。由于Laplace算子的出现,线性化方程对应的特征方程变为可列个超越方程,而每个超越方程都产生可列个Hopf分支临界值??,一般情况下很难判断这些临界值的大小顺序。本文给出了在扩散系数满足一定条件时对应于Laplace算子有限个特征值的临界值??的顺序。并且分析了空间齐次和非齐次的Hopf分支周期解的性质。
The understanding of patterns and mechanisms of spatial dispersal of interactingspecies is an issue of significant current interest in conservation biology and ecology, andbiochemical reactions. A typical type of interaction is the one between a pair of predatorand prey, or more generally, a pair of consumer and resource. Mathematical models of theinteraction can be written as reaction-diffusion equations, ordinary differential equations,and functional differential equations, which are important in the research of nonlineardifferential equations. The methods of our studies include the classical dynamical systemtheory, analytical semigroup and topological methods, etc.
For the predator-prey system with strong Allee effect in prey, the dynamical prop-erties of some special systems have only been obtained by numerical simulations. Thispaper aims to the complete analytical analysis. Unlike the Logistic growth, the strongAllee effect induces bistability, for which currently known methods cannot be appliedto, this paper greatly improves some methods and technical constructions. Details are asfollows:
1. A completely global bistability analysis has been obtained for the system of ordi-nary differential equations, which is independent of spatial variables. By taking a compo-nent of positive coexistence equilibrium as parameter, two global bifurcation values aregiven: heteroclinic loop bifurcation point and Hopf bifurcation point. When the parameterchanges, the global stable zero equilibrium, the unique heteroclinic loop, the unique limitcycle as an alternative stable state, the coexistence equilibrium point as alternative stablestate are obtained successively. In addition, an improved Dulac criteria is given whenproving the nonexistence of periodic solution. Also a predator-prey systems with Alleeeffect could have even richer dynamic structure like multiple limit cycles. Our rigorousanalysis can be applied to most existing predator-prey models with strong Allee effect.Our analytical analysis are mostly new and complete, and the results do not depend onthe specific algebraic forms or parametrization of the nonlinear functions in the models.The analysis also provide new method and approach for the dynamical behavior of otherplanar systems.
2. By constructing super-sub solutions for predator-prey system with strong Allee effect, we obtain the basic dynamical behavior, including the existence of global solutionand asymptotic behavior. Also the a priori estimate bounds of solutions and the spatial ho-mogeneous and nonhomogeneous periodic solutions are given. Especially, a large enoughinitial predator population will always lead to the extinction, i.e. the convergence to thesteady state (0, 0), which implies that (0, 0) is always a locally stable steady state withbasin of attraction including all large ??0 for a given ??0. Thus the system is bistable (ormulti-stable) if there is another locally stable steady state solution or periodic orbit. More-over, if the amount of initial prey population is less than the threshold value, (0, 0) willalways is globally asymptotically stable, which is also the characteristic of strong Alleeeffect. All the results show that the strong Allee effect essentially enlarges the complexityof spatial-tempo behavior of predator-prey reaction-diffusion systems.
3. The elliptic system independent of time is analyzed in detail. A priori estimatesand nonexistence of positive steady state solutions are obtained. The stability and bifur-cation of semi-trivial steady state solution are given thorough. It is difficult to obtain thepositive lower bound of positive steady state solutions since the bistability deduced bystrong Allee effect and the semitrivial steady state solutions, the classical Leray-Schauderdegree cannot be applied to obtain the nonconstant positive steady state solutions. There-fore we make use of the global bifurcation theory generalized by Shi and Wang[1] to getthe nonconstant positive steady state solutions, i.e. spatial pattern formation.
4. The stability and bifurcation analysis of the functional and partial functional dif-ferential equations with two discrete delays are considered. The delays will increase theinstability of predator-prey system with strong Allee effect. Based on this, the in?uenceof diffusion is considered. Since the appearance of Laplace operator, the characteristicequation of linearization equals to a sequence of countably many transcendental equa-tions. While each transcendental equation may generate countable Hopf bifurcation criti-cal value ?? , which makes the ordering of the critical value sequence difficult. The paperclarifies the sequence of critical value ?? corresponding to eigenvalues of the Laplace oper-ator. Moreover the spatial homogeneous and nonhomogeneous Hopf bifurcation periodicsolutions and their properties are analyzed.
当种群具有强Allee效应增长时,此前只有数值上的模拟结果,本文主要对具有强Allee效应的捕食-食饵系统进行了系统的解析分析。不同于Logistic型增长的捕食-食饵系统,强Allee效应使系统具有双稳定性结构,因此很多常用的研究方法用起来存在困难或者不再适用,需要一些改进的方法及一定的构造性技巧,本文主要工作如下:
1.对常微分系统进行了详细的全局双稳定动力学分析,以正平衡点的第一个分量为参数,得到了两个全局分支值:异宿轨道环分支值和Hopf分支值。在参数不同的取值范围内分别得到了相应的全局稳定的零平衡点、唯一的异宿轨道环、全局双稳定的零平衡点和周期轨道、全局双稳定的零平衡点和正平衡点及全局双稳定的零平衡点和半平衡点。在证明周期解的不存在性时给出了改进的Dulac函数判断方法,这个结果在具体应用时更为适用。此外,本文还证明了具有其他类型Allee效应的一些捕食-食饵系统还有更丰富的性质,例如多个周期轨道。本文严格的分析可以应用到很多具体的强Allee效应的捕食-食饵模型。其解析结果几乎是最新,最完整的,且不依赖于非线性函数具体的代数形式,并对其它二维常微分方程组的动力学行为分析提供了系统的方法和途径。
2.通过构造上、下解,能量估计等方法研究了具强Allee效应的反应扩散方程组的基本动力学行为,得到了整体解的存在性及渐近性,给出了解的先验估计,并得到了系统的双稳定性及空间齐次和非齐次的周期解。特别地,当捕食者的初始值足够大时,解最终趋于(0,0),这也表明(0,0)总是局部稳定的,即给定食饵的初始值, (0,0)的吸引域包含所有充分大的.因此若存在其他的局部稳定的稳态解或者周期解,则系统是双稳定(或者多重稳定)的;当食饵的初始值小于强效应的门槛值时, (0,0)也是全局渐近稳定的,这是具有强Allee效应的捕食-食饵系统的特征。这些结果也表明强Allee效应从本质上增加了反应扩散方程组时空动力学行为的复杂性。
3.对具有强Allee效应的椭圆方程组进行了详细的分析。利用椭圆方程的正则性估计得到了非常数正稳态解的先验估计及不存在性;分析了平凡稳态解的稳定性和半平凡稳态解的分支情况;由于强Allee效应下的双稳定结构,且系统有非常多的半平凡稳态解分支,因而难以证得稳态解的正下界,常用的Leray-Schauder度理论不能够得到非常数的正稳态解,本文采用的是史峻平和王学峰[1]推广的全局分支理论得到非常数正稳态解的存在性,即空间的模式生成;
4.分别考虑带两个离散滞量的泛函和偏泛函微分方程组的稳定性和分支分析。讨论了滞量对强Allee效应捕食-食饵系统的不稳定性影响,及在此基础上扩散项对系统稳定性的影响。由于Laplace算子的出现,线性化方程对应的特征方程变为可列个超越方程,而每个超越方程都产生可列个Hopf分支临界值??,一般情况下很难判断这些临界值的大小顺序。本文给出了在扩散系数满足一定条件时对应于Laplace算子有限个特征值的临界值??的顺序。并且分析了空间齐次和非齐次的Hopf分支周期解的性质。
The understanding of patterns and mechanisms of spatial dispersal of interactingspecies is an issue of significant current interest in conservation biology and ecology, andbiochemical reactions. A typical type of interaction is the one between a pair of predatorand prey, or more generally, a pair of consumer and resource. Mathematical models of theinteraction can be written as reaction-diffusion equations, ordinary differential equations,and functional differential equations, which are important in the research of nonlineardifferential equations. The methods of our studies include the classical dynamical systemtheory, analytical semigroup and topological methods, etc.
For the predator-prey system with strong Allee effect in prey, the dynamical prop-erties of some special systems have only been obtained by numerical simulations. Thispaper aims to the complete analytical analysis. Unlike the Logistic growth, the strongAllee effect induces bistability, for which currently known methods cannot be appliedto, this paper greatly improves some methods and technical constructions. Details are asfollows:
1. A completely global bistability analysis has been obtained for the system of ordi-nary differential equations, which is independent of spatial variables. By taking a compo-nent of positive coexistence equilibrium as parameter, two global bifurcation values aregiven: heteroclinic loop bifurcation point and Hopf bifurcation point. When the parameterchanges, the global stable zero equilibrium, the unique heteroclinic loop, the unique limitcycle as an alternative stable state, the coexistence equilibrium point as alternative stablestate are obtained successively. In addition, an improved Dulac criteria is given whenproving the nonexistence of periodic solution. Also a predator-prey systems with Alleeeffect could have even richer dynamic structure like multiple limit cycles. Our rigorousanalysis can be applied to most existing predator-prey models with strong Allee effect.Our analytical analysis are mostly new and complete, and the results do not depend onthe specific algebraic forms or parametrization of the nonlinear functions in the models.The analysis also provide new method and approach for the dynamical behavior of otherplanar systems.
2. By constructing super-sub solutions for predator-prey system with strong Allee effect, we obtain the basic dynamical behavior, including the existence of global solutionand asymptotic behavior. Also the a priori estimate bounds of solutions and the spatial ho-mogeneous and nonhomogeneous periodic solutions are given. Especially, a large enoughinitial predator population will always lead to the extinction, i.e. the convergence to thesteady state (0, 0), which implies that (0, 0) is always a locally stable steady state withbasin of attraction including all large ??0 for a given ??0. Thus the system is bistable (ormulti-stable) if there is another locally stable steady state solution or periodic orbit. More-over, if the amount of initial prey population is less than the threshold value, (0, 0) willalways is globally asymptotically stable, which is also the characteristic of strong Alleeeffect. All the results show that the strong Allee effect essentially enlarges the complexityof spatial-tempo behavior of predator-prey reaction-diffusion systems.
3. The elliptic system independent of time is analyzed in detail. A priori estimatesand nonexistence of positive steady state solutions are obtained. The stability and bifur-cation of semi-trivial steady state solution are given thorough. It is difficult to obtain thepositive lower bound of positive steady state solutions since the bistability deduced bystrong Allee effect and the semitrivial steady state solutions, the classical Leray-Schauderdegree cannot be applied to obtain the nonconstant positive steady state solutions. There-fore we make use of the global bifurcation theory generalized by Shi and Wang[1] to getthe nonconstant positive steady state solutions, i.e. spatial pattern formation.
4. The stability and bifurcation analysis of the functional and partial functional dif-ferential equations with two discrete delays are considered. The delays will increase theinstability of predator-prey system with strong Allee effect. Based on this, the in?uenceof diffusion is considered. Since the appearance of Laplace operator, the characteristicequation of linearization equals to a sequence of countably many transcendental equa-tions. While each transcendental equation may generate countable Hopf bifurcation criti-cal value ?? , which makes the ordering of the critical value sequence difficult. The paperclarifies the sequence of critical value ?? corresponding to eigenvalues of the Laplace oper-ator. Moreover the spatial homogeneous and nonhomogeneous Hopf bifurcation periodicsolutions and their properties are analyzed.
引文
1 J. P. Shi, X. F. Wang. On the Global Bifurcation for Quasilinear Elliptic Systemson Bounded Domains[J]. J. Diff. Equ., 2009, 246:2788–2812.
2 W. C. Allee. Animal Aggregations: A Study in General Sociology[M]. Universityof Chicago Press, 1931.
3 F. Courchamp, L. Berec, J. Gascoigne. Allee Effects in Ecology and Conserva-tion[M]. University of Chicago Press, 2008.
4 B. Dennis. Allee Effect: Population Growth, Critical Density and Chance of Ex-tinction[J]. Natur. Resource Modeling., 1989, 3:481–538.
5 J. P. Shi, R. Shivaji. Persistence in Reaction Diffusion Models with Weak AlleeEffect[J]. J. Math. Biol., 2006, 52:807–829.
6 M. E. Wang, M. Kot. Speeds of Invasion in a Model with Strong Or Weak AlleeEffects[J]. Math. Biosci., 2001, 171:83–97.
7 L. Berec, E. Angulo, F. Courchamp. Multiple Allee Effects and Population Man-agement[J]. Trends in Ecol. Evol., 2007, 22:185–191.
8 M. R. Owen, M. A. Lewis. How Predation Can Slow, Stop Or Reverse a PreyInvasion[J]. Bull. Math. Biol., 2001, 63:655–684.
9 S. Petrovskii, A. Morozov, E. Venturinoi. Allee Effect Makes Possible Patchy In-vasion in a Predator-prey System[J]. Ecol. Lett., 2002, 5:345–352.
10 R. Lande. Extinction Thresholds in Demographic Models of Territorial Popula-tions[J]. Amer. Natur., 1987, 130:624–635.
11 M. A. Burgman, S. Ferson, H. R. Akcakaya. Risk Assessment in ConservationBiology[M]. Chapman and Hall, London, 1993.
12 P. A. Stephens, W. J. Sutherland. Consequences of the Allee Effect for Behaviour,Ecology and Conservation[J]. Trends in Ecol. Evol., 1999, 14:401–405.
13 J. C. Gascoigne, R. N. Lipcius. Allee Effects Driven by Predation[J]. J. Appl. Ecol.,2004, 41:801–810.
14 J. F. Jiang, J. P. Shi. Bistability Dynamics in some Structured Ecological Models,in“Spatial Ecology”[M]. CRC Press, 2009.
15 P. A. Stephens, W. J. Sutherland, R. P. Freckleton. What Is the Allee Effect[J].Oikos, 1999, 87:1851–1890.
16 H. L. Smith. Monotone Dynamical System (an Introduction to the Theory of Com-petitive and Cooperative Systems)[M]. AMS., 1995.
17 X. Q. Zhao. Dynamical Systems in Population Biology[M]. Springer, 2003.
18 R. Fisher. The Wave of Advance of Advantageous Genes[J]. Annals of Eugenics,1937, 7:355–369.
19 A. Kolmogorov, I. Petrovsy, N. Piskounov. Study of the Diffusion Equation withGrowth of the Quantity of Matter and its Applications to a Biological Problem[J].Bull. Math, 1937, 1:1–25.
20 J. G. Skellam. Random Dispersal in Theoretical Populations[J]. J. Diff. Equ., 1951,38:196–218.
21 H. Kierstead, L. B. Slobodkin. The Size of Water Masses Containing PlanktonBlooms[J]. J. Mar. Res, 1953, 12:141–147.
22 A. M. Turing. The Chemical Basis of Morphogenesis[J]. Phil. Trans. R. Soc.London Ser. B, 1952, 237:37–72.
23 A. J. Lotka. Elements of Physical Biology[M]. Williams and Wilkins, Baltimore,1925.
24 V. Volterra. Fluctuations in the Abundance of Species[J]. Nature, 1926, 118:558.
25 C. S. Holling. The Components of Predation as Revealed by a Study of SmallMammal Predation of the European Pine Saw?y[J]. Canadian Entomologist., 91:293–320.
26 V. S. Ivlev. Experimental Ecology of the Feeding of Fishes[M]. Yale UniversityPress, 1955.
27 N. Kazarinov, P. van den Driessche. A Model Predator-prey Systems with Func-tional Rresponse[J]. Oecologia, 1978, 39:125–134.
28 P. Turchin. Complex Population Dynamics: A Theoretical/empirical Synthesis[M].Princeton University Press, 2003.
29 K. S. Cheng. Uniqueness of a Limit Cycle for a Predator-prey System[J]. SIAM J.Math. Anal., 1981, 12:541–548.
30 S. B. Hsu. On Global Stability of a Predator-prey System[J]. Math. Biosci., 1978,39:1–10.
31 S. B. Hsu, J. P. Shi. Relaxation Oscillator Profile of Limit Cycle in Predator-preySystem[J]. Discrete Contin. Dynam. Systems. Ser. B, 2009, 11:893–931.
32 Y. Kuang, H. I. Freedman. Uniqueness of Limit Cycles in Gause-type Models ofPredator-prey Systems[J]. Math. Biosci., 1988, 88:67–84.
33 R. M. May. Limit Cycles in Predator-prey Communities[J]. Science, 1972,177:900–902.
34 M. L. Rosenzweig. Paradox of Enrichment: Destabilization of Exploitation Ecosys-tems in Ecological Time[J]. Science, 1971, 171:385–387.
35 D. Xiao, Z. Zhang. On the Uniqueness and Nonexistence of Limit Cycles forPredator-prey System[J]. Nonlinearity, 2003, 16:1–17.
36 M. L. Rosenzweig, R. MacArthur. Graphical Representation and Stability Condi-tions of Predator-prey Interactions[J]. Amer. Natur., 1963, 97:209–223.
37 F. Albrecht, H. Gatzke, A. Haddad, et al. The Dynamics of Two Interacting Popu-lations[J]. J. Math. Anal. Appl., 1974, 46:658–670.
38 F. Albrecht, H. Gatzke, N. Wax. Stable Limit Cycles in Prey-predator Popula-tions[J]. Science, 1973, 181:1074–1075.
39 G. A. K. van Voorn, L. Hemerik, M. P. Boer, et al. Heteroclinic Orbits IndicateOverexploitation in Predator Prey Systems with a Strong Allee Effect[J]. Math.Biosci., 2007, 209:451–469.
40 N. D. Alikakos. An Application of the Invarianceprinciple to Reaction-diffusionEquations[J]. J. Diff. Equ., 1979, 33:201–225.
41 E. D. Conway, D. Hoff, J. A. Smoller. Large Time Behavior of Solutions of Systemsof Nonlinear Reaction-diffusion Equations[J]. SIAM J. Appl. Math., 1978, 35:1–16.
42 R. S. Cantrell, C. Cosner. Spatial Ecology via Reaction-diffusion Equations, WileySeries in Mathematical and Computational Biology[M]. John Wiley & Sons, Ltd.,Chichester, 2003.
43 C. V. Pao. Nonlinear Parabolic and Elliptic Equations[M]. Plenum Press, NewYork, 1992.
44 J. Smoller. Shock Waves and Reaction-diffusion Equations[M]. Springer-Verlag,New York, 1994.
45叶其孝.反应扩散方程[M].科学出版社, 1994.
46 K. N. Chueh, C. C. Conley, J. A. Smoller. Positively Invariant Regions for Systemsof Nonlinear Diffusion Equations[J]. Indiana Univ. Math., 1983, 17:217–234.
47 N. D. Alikakos. A Lyapunov Functional for a Class of Reaction-diffusion Systems.Modeling and Differential Equations in Biology. Lecture Notes in Pure and Appl.Math[J]. 1980:153–170.
48 P. Demotoni, F. Rothe. Convergence to Homogeneous Equilibrium State for Gen-eralized Volterra-lotka Systems[J]. SIAM J. Appl. Math., 1979, 37:648–663.
49王明新.非线性椭圆方程[M].科学出版社, 2008.
50 F. Q. Yi, J. J. Wei, J. P. Shi. Bifucation Analysis of a Diffusive Predator-prey Systemwith Holling Type-ii Function Response[J]. J. Diff. Equ., 2009, 246:1944–1977.
51 E. D. Conway, D. Hoff, J. A. Smoller. Stability and Bifurcation of Steady StateSolutions for Predator Prey Equations[J]. Adv. in Appl. Math., 1982, 3:288–334.
52 C. V. Pao. On Nonlinear Reaction-diffusion Systems[J]. J. Math. Anal. Appl., 1982,87:165–198.
53 P. Koman, A. W. Leung. A General Monotone Scheme for Elliptic Systems withApplications to Ecological Models[J]. Proc. Roy. Soc. Edin. A, 1986, 102:315–325.
54 J. Blat, K. J. Brown. Bifurcation of Steady State Solutions on Predator Prey andCompetition Systems[J]. Proc. Roy. Soc. Edin. A, 1984, 97:21–34.
55 J. Blat, K. J. Brown. Global Bifurcation of Positive Solutions in some Systems ofElliptic Equations[J]. SIAM. J. Math. Anal., 1986, 17:1339–1353.
56 E. N. Dancer. On Positive Solutions of some Paris of Differential Equations, I[J].Trans. Amer. Math. Soc., 1984, 284:729–743.
57 E. N. Dancer. On Positive Solutions of some Paris of Differential Equations, Ii[J].J. Diff. Equ., 1985, 60:236–258.
58 L. Li. Coexistence Theorem of Steady State for Predator Prey Systems[J]. Trans.Amer. Math. Soc., 1988, 305:143–166.
59 L. Li. On the Uniqueness and Ordering of Steady State of Predator Prey Systems[J].Proc. Roy. Soc. Edin. A, 1988, 110:295–303.
60 E. N. Dancer. On the Existence and Uniqueness of Positive Solutions for CompetingSpecies Models with Diffusion[J]. Tran. Amer. Math. Soc., 1991, 326:829–859.
61 E. N. Dancer. On Uniqueness and Stability for Solutions of Singularly PerturbedPredator Prey Type Equations with Diffusion[J]. J. Diff. Equ., 1993, 102:1–32.
62 Y. Lou, W. M. Ni. Diffusion vs Cross-diffusion: An Elliptic Approach[J]. J. Diff.Equ., 1999, 154:157–190.
63 Y. H. Du, Y. Lou. Some Uniqueness and Exact Multiplicity Results for a PredatorPrey Model[J]. Proc. Roy. Soc. Edin. A, 2001, 131:321–349.
64 Y. H. Du, Y. Lou. Qualitive Behavior of Positive Solutions of a Predator PreyModel: Effects of Saturation[J]. Tran. Amer. Math, Soc., 1997, 349:2443–2475.
65 R. Peng, M. X. Wang. Positive Steady States of the Holling-tanner Predator PreyModel with Diffusion[J]. Proc. Roy. Soc. Edin. A, 2005, 135:149–164.
66 H. Matano. Asymptotic Behavior and Stability of Solutions of Semilinear DiffusionEquations[J]. Publ. Res. Inst. Math. Sci., 1979, 15:401–454.
67 H. Matano, M. Mimura. Pattern Formations in Competition-diffusion Systems inNonconvex Domains[J]. Publ. Res. Inst. Math. Sci., 1983, 19:1049–1079.
68 J. D. Murray. Nonexistence of Wave Solutions for the Class of Reaction-diffusionEquations Given by the Volterra Interaction Population Equations with Diffu-sion[J]. J. Theor. Biol., 1975, 52:259–469.
69 L. A. Segel, J. L. Jackson. Dissipative Structure: An Explaination and an EcologicalExample[J]. J. Theor. Biol., 1975, 37:545–559.
70 L. A. Segel, S. A. Levin. Application of Nonlinear Stability Theory to the Study ofthe Effects of Diffusion on Predator-prey Interactions[J]. AIP Conf. Proc., Amer.Inst. Phys., 1976, 27:123–156.
71 V. Volterra. Lecons Sur La Th?′? Orie Math?′?matique De La Lutte Pour La Vie[M].Gauthiers-Villars, Paris, 1931.
72 L. S. Dai. Nonconstant Periodic Solutions in Predator-prey Systems with Continu-ous Time Delay[J]. Math. Biosci., 1981, 53:149–157.
73 T. Faria. Stability and Bifurcation for a Delayed Predator Prey Model and the Effectof Diffusion[J]. J. Math. Anal. Appl, 2001, 254:433–463.
74 Y. Kuang. Delay Differential Equations with Applications in Population Dynam-ics[M]. Academic Press, New York, 1993.
75 J. J. Wei, M. Y. Li. Global Existence of Periodic Solutions in Tri-neuron NetworkModel with Delays[J]. Phy. D, 2004, 198:106–119.
76 J. J. Wei, S. Ruan. Stability and Bifurcation in a Neural Network Model with TwoDelays[J]. Phy. D, 1999, 130:225–272.
77 J. H. Wu. Theory and Application of Partial Functional Differential Equaitons[M].Spring-Verlag, NewYork, 1989.
78 J. H. Wu. Symmetric Functional Differential Equations and Neural Networks withMemory[J]. Trans. Amer. Math. Soc., 1998, 350:4799–4838.
79 X. Lin, J. W. H. So, J. H. Wu. Center Manifolds for Partial Differential Equationswith Delays[J]. Proc. Roy. Soc. Edin. A, 1992, 122:237–254.
80 T. Faria. Normal Forms and Hopf Bifurcation for Partial Differential Equationswith Delays[J]. Trans. Amer. Math. Soc., 2000, 352:2217–2238.
81 T. Faria, L. T. Magalhaes. Normal Form for Retarded Functional Equations withParameters and Applications to Hopf Bifurcation[J]. J. Diff. Equ., 1995, 122:181–200.
82 T. Faria, L. T. Magalhaes. Normal Form for Retarded Functional Equations andApplications to Bogdanov-takens Singularity[J]. J. Diff. Equ., 1995, 122:201–224.
83 Y. Su, A. Y. Wan, J. J. Wei. Bifurcation Analysis in a Diffusive Food-limite Modelwith Time Delay[J]. Appl. Anal., 2010, 89:1161–1181.
84 Y. Su, J. J. Wei, J. P. Shi. Hopf Bifurcations in a Reaction-diffusion PopulationModel with Delay Effect[J]. J. Diff. Equ., 2009, 247:1156–1184.
85 X. P. Yan, W. T. Li. Stability and Hopf Bifurcation for a Delayed CooperativeSystem with Diffusion Effects[J]. Int. J. Bifur. Chaos., 2008, 18:441–453.
86 M. A. Lewis, P. Kareiva. Allee Dynamics and the Spread of Invading Organisms[J].Theor. Popul. Biol., 1993, 43:141–158.
87 A. Morozov, S. Petrovskii, B. L. Li. Bifurcations and Chaos in a Predator-preySystem with the Allee Effect[J]. Proc. Roy. Soc. London Series B-Biol. Sci., 2004,271:1407–1414.
88 A. Morozov, S. Petrovskii, B. L. Li. Spatiotemporal Complexity of Patchy Invasionin a Predator-prey System with the Allee Effect[J]. J. Theor. Biol., 2006, 238:18–35.
89 S. D. Boukal, W. M. Sabelis, L. Berec. How Predator Functional Responses andAllee Effects in Prey Effect the Paradox of Enrichment and Population Collapses[J].Theor. Popul. Biol., 2007, 72:136–147.
90 H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology andEpidemiology. Theory, Models and Simulation. Chapman Hall/crc Mathematicaland Computational Biology Series[M]. Chapman Hall/CRC, Boca Raton, FL, 2008.
91 S. Petrovskii, A. Morozov, B. L. Li. Regimes of Biological Invasion in a Predator-prey System with the Allee Effect[J]. Bull. Math. Biol., 2005, 67:637–661.
92 A. D. Bazykin. Nonlinear Dynamics of Interacting Populations. World ScientificSeries on Nonlinear Science. Series A: Monographs and Treatises, 11[M]. WorldScientific Publishing, 1998.
93 A. Pablo, G. Eduardo, S. Eduardo. Three Limit Cycles in a Leslie-gower Predator-prey Model with Additive Allee Effect[J]. SIAM J. Appl. Math., 2009, 69:1244–1262.
94 E. Gonza′lez-Olivares, B. Gonza′lez-Yavz, E. Sa′ez, et al. On the Number of LimitCycles in a Predator Prey Model with Non-monotonic Functional Response[J]. Dis-crete Contin. Dynam. Systems. Ser. B, 2006, 6:525–534.
95 S. Ruan, D. Xiao. Global Analysis in a Predator-prey System with NonmonotonicFunctional Response[J]. SIAM J. Appl. Math., 2000, 61:1445–1472.
96 R. Arditi, L. R. Ginzburg. Coupling in Predator-prey Dynamics: Ratio-dependence[J]. J. Theor. Biol., 1989, 139:311–326.
97 S. B. Hsu, T. W. Hwang, Y. Kuang. Global Analysis of the Michaelis-menten-typeRatio-dependent Predator-prey System[J]. J. Math. Biol., 2001, 42:489–506.
98 Y. Kuang, E. Beretta. Global Qualitative Analysis of a Ratio-dependent Predator-prey System[J]. J. Math. Biol., 1998, 36:389–406.
99 D. Xiao, S. Ruan. Global Gynamics of a Ratio-dependent Predator-prey System[J].J. Math. Biol., 2001, 43:268–290.
100 S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M].Springer-Verlag, New York, 1990.
101 S. B. Hsu. Ordinary Differential Equations with Applications[M]. Series on Ap-plied Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,2006.
102 E. D. Conway, J. A. Smoller. Global Analysis of a System of Predator-prey Equa-tions[J]. SIAM J. Appl. Math., 1986, 46:630–642.
103 B. D. Hassard, N. D. Kazarinoff, Y. Wan. Theory and Applications of Hopf Bi-furcation[J]. London Math. Soc. Lecture Note Ser. Cambridge University Press,Cambridge, MA, 1981, 41:539–547.
104 Y. A. Kuznetsov. Elements of Applied Bifurcation Theory[M]. Springer-Verlag,New York, 2004.
105 M. L. Rosenzweig. Why the Prey Curve Has a Hump[J]. Amer. Natur., 1969,103:81–87.
106 D. Xiao, Z. Zhang. On the Existence and Uniqueness of Limit Cycles for General-ized Li?′?nard Systems[J]. J. Math. Anal. Appl., 2008, 343:299–309.
107 X. Zeng, Z. Zhang, S. Gao. On the Uniqueness of the Limit Cycle of the General-ized Lie′nard Equation[J]. Bull. London Math. Soc., 1994, 26:213–247.
108 Z. F. Zhang. Proof of the Uniqueness Theorem of Limit Cycles of GeneralizedLie′nard Equations[J]. Appl. Anal., 1986, 23:63–76.
109 J. Hofbauer, J. So. Multiple Limit Cycles for Predator-prey Models[J]. Math.Biosci., 1990, 99:71–75.
110 Y. Kuang. Global Stability of Gause-type Predator-prey System[J]. J. Math. Biol.,1990, 28:462–474.
111 Y. P. Liu. Geometric Criteria for the Nonexistence of Cycles in Gause-typePredator-prey Systems[J]. Proc. Amer. Math. Soc., 2005, 133:3619–3626.
112 J. C. Alexander, J. A. Yorke. Global Bifurcations of Periodic Orbits[J]. Amer. J.Math., 1978, 100:263–292.
113 S. N. Chow, J. Mallet-Paret. The Fuller Index and Global Hopf Bifurcation[J]. J.Diff. Equ., 1978, 29:66–85.
114 J. A. Polking, D. Arnold. Ordinary Differential Equations Using Matlab. 3rd Edi-tion[M]. Prentice Hall, 2003.
115 A. Dhooge, W. Govaerts, Y. A. Kuznetsov. Matcont: A Matlab Package for Numer-ical Bifurcation Analysis of Odes[J]. ACM Trans. Math. Software, 2003, 29:141–164.
116 H. R. Thieme, T. Dhirasakdanon, Z. Han, et al. Species Decline and Extinction:Synergy of Infectious Disease and Allee Effect[J]. J. Biol. Dyna., 2009, 3:305–323.
117 F. M. Hilker, M. Langlais, H. Malchow. The Allee Effect and Infectious Diseases:Extinction, Multistability, and the (dis-) Appearance of Oscillations[J]. Amer.Natur., 2009, 173:72–88.
118 S. D. Boukal, L. Berec. Single-species Models of the Allee Effect: ExtinctionBoundaries, Sex Ratios and Mate Encounters[J]. J. Theor. Biol., 2002, 218:375–394.
119 Y. Gruntfest, R. Arditi, Y. Dombrovsky. A Fragmented Population in a VaryingEnvironment[J]. J. Theor. Biol., 1997, 185:539–547.
120 E. O. Wilson, W. H. Bossert. A Primer of Population Biology[M]. MA, U.S.A.,1971.
121 Y. Takeuchi. Global Dynamical Properties of Lotka-volterra Systems[M]. WorldScientific Publishing Company, 1996.
122 J. Jacobs. Cooperation, Optimal Density and Low Density Thresholds: Yet AnotherModification of the Logistic Model[J]. Oecologia, 1984, 64:395–398.
123 S. R. Zhou, Y. F. Liu, G. Wang. The Stability of Predator-prey Systems Subject tothe Allee Effects[J]. Theor. Popul. Biol., 2005, 67:23–31.
124 R. M. Nisbet, W. S. C. Gurney. Modelling Fluctuating Populations[M]. Chichester.,1982.
125 Y. H. Du, J. P. Shi. Some Recent Results on Diffusive Predator-prey Models inSpatially Heterogeneous Environment[J]. Fields Inst. Commun. Amer. Math. Soc.,Providence, RI, 2006, 48:141–164.
126 Y. H. Du, J. P. Shi. Allee Effect and Bistability in a Spatially HeterogeneousPredator-prey Model[J]. Trans. Amer. Math. Soc., 2007, 359:4557–4593.
127 J. H. Huang, G. Lu, S. G. Ruan. Existence of Traveling Wave Solutions in a Diffu-sive Predator-prey Model[J]. J. Math. Biol., 2003, 46:132–152.
128 W. Ko, K. Ryu. Qualitative Analysis of a Predator-prey Model with Holling Type IiFunctional Response Incorporating a Prey Refuge[J]. J. Diff. Equ., 2006, 231:534–550.
129 Y. Lou, W. M. Ni. Diffusion, Self-diffusion and Cross-diffusion[J]. J. Diff. Equ.,1996, 131:79–131.
130 X. Y. Chen, P. Pola′cˇik. Gradient-like Structure and Morse Decompositions forTime-periodic One-dimensional Parabolic Equations[J]. J. Dynam. Diff. Equ.,1995, 7:73–107.
131 J. F. Jiang, J. P. Shi. Dynamics of a Reaction-diffusion System of AutocatalyticChemical Reaction[J]. Discrete Contin. Dynam. System., 2008, 21:254–258.
132 K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically Autonomous Semi?ows:Chain Recurrence and Lyapunov Functions[J]. Trans. Amer. Math. Soc., 1995,347:1669–1685.
133 R. S. Cantrell, C. Cosner, V. Hutson. Permanence in Ecological Systems with Spa-tial Heterogeneity[J]. Proc. Roy. Soc. Edin. A, 1993, 123:533–559.
134 S. L. Hollis, R. H. Martin, M. Pierre. Global Existence and Boundedness inReaction-diffusion Systems[J]. SIAM J. Math. Anal., 1987, 18:744–761.
135 D. Henry. Geometric Theory of Eemilinear Parabolic Equations, Lecture Notes inMathematics[M]. Springer-Verlag, Berlin-New York, 1981.
136 R. Peng, J. P. Shi, M. X. Wang. On Stationary Patterns of a Reaction-diffusionModel with Autocalysis and Saturation Law[J]. Nonlinearity, 2008, 21:1471–1488.
137 J. P. Shi. Solution Set of Semilinear Elliptic Equations, Global Bifurcation andExact Multiplicity[M]. World Scientific Publishing, 2010.
138 P. H. Rabinowitz. Minimax Methods in Critical Point Theorey with Applicationsto Differential Equaitons[M]. Conference Board of the Mathematical Sciences Re-gional Conference Series in Mathematics., 1984.
139 P. H. Rabinowitz. Some Global Results for Nonlinear Eigenvalue Problems[J]. J.Funct. Anal., 1971, 7:487–513.
140 M. G. Crandall, P. H. Rabinowitz. Bifurcation from Simple Eigenvalues[J]. J.Funct. Anal., 1971, 8:321–340.
141 J. P. Shi. Semilinear Neumann Boundary Value Problems on a Rectangle[J]. Trans.Amer. Math. Soc., 2002, 354:3117–3154.
142 J. Smoller, A. Wasserman. Global Bifurcation of Steady-state Solutions[J]. J. Diff.Equ., 1981, 39(2):269–290.
143 J. F. Jiang, X. Liang, X. Q. Zhao. Saddle-point Behavior for Monotone Semi?owsand Reaction-diffusion Models[J]. J. Diff. Equ., 2004, 203:313–330.
144 J. Y. Jin, J. P. Shi, J. J. Wei, et al. Patterned Solutions in Diffusive Lengyel-epsteinSystem of Cima Chemical Reaction[J]. Rocky Mountain Jour. Math, 2011, ToAppear.
145 Z. Liu, R. Yuan. The Effect of Diffusion for a Predator Prey System with Non-monotonic Functional Response[J]. Int. J. Bifur. Chaos., 2004, 12:4309–4316.
146 K. Cooke, Z. Grossman. Discrete Delay, Distributed Delay and StabilitySwitches[J]. J. Math. Anal. Appl., 1982, 86:592–672.
147 J. K. Hale. Theory of Functional Differential Equations[M]. Springer, New York,1977.
148 S. N. Chow, J. K. Hale. Method of Bifurcation Theory[M]. Springer-Verlag, NewYork, 1982.
149 E. Beretta, Y. Kuang. Geometric Stability Switch Criteria in Delay Differential Sys-tem with Delay Dependent Parameters[J]. SIAM. J. Math. Anal., 2003, 33:1144–1165.
150 Y. L. Song, J. J. Wei. Local Hopf Bifurcation and Global Periodic Solutions in aDelayed Predator Prey System[J]. J. Math. Anal. Appl., 2005, 301:1–21.
151 Y. H. Du, J. P. Shi. A Diffusive Predator-prey Model with a Protection Zone[J]. J.Diff. Equ., 2006, 229:63–91.
152 R. Peng, J. P. Shi, M. X. Wang. Stationary Pattern of a Ratio-dependent Food ChainModel with Diffusion[J]. SIAM J. Appl. Math., 2007, 65:1479–1503.
153 M. X. Wang. Non-constant Positive Steady States of the Sel’kov Model[J]. J. Diff.Equ., 2003, 190:600–620.
2 W. C. Allee. Animal Aggregations: A Study in General Sociology[M]. Universityof Chicago Press, 1931.
3 F. Courchamp, L. Berec, J. Gascoigne. Allee Effects in Ecology and Conserva-tion[M]. University of Chicago Press, 2008.
4 B. Dennis. Allee Effect: Population Growth, Critical Density and Chance of Ex-tinction[J]. Natur. Resource Modeling., 1989, 3:481–538.
5 J. P. Shi, R. Shivaji. Persistence in Reaction Diffusion Models with Weak AlleeEffect[J]. J. Math. Biol., 2006, 52:807–829.
6 M. E. Wang, M. Kot. Speeds of Invasion in a Model with Strong Or Weak AlleeEffects[J]. Math. Biosci., 2001, 171:83–97.
7 L. Berec, E. Angulo, F. Courchamp. Multiple Allee Effects and Population Man-agement[J]. Trends in Ecol. Evol., 2007, 22:185–191.
8 M. R. Owen, M. A. Lewis. How Predation Can Slow, Stop Or Reverse a PreyInvasion[J]. Bull. Math. Biol., 2001, 63:655–684.
9 S. Petrovskii, A. Morozov, E. Venturinoi. Allee Effect Makes Possible Patchy In-vasion in a Predator-prey System[J]. Ecol. Lett., 2002, 5:345–352.
10 R. Lande. Extinction Thresholds in Demographic Models of Territorial Popula-tions[J]. Amer. Natur., 1987, 130:624–635.
11 M. A. Burgman, S. Ferson, H. R. Akcakaya. Risk Assessment in ConservationBiology[M]. Chapman and Hall, London, 1993.
12 P. A. Stephens, W. J. Sutherland. Consequences of the Allee Effect for Behaviour,Ecology and Conservation[J]. Trends in Ecol. Evol., 1999, 14:401–405.
13 J. C. Gascoigne, R. N. Lipcius. Allee Effects Driven by Predation[J]. J. Appl. Ecol.,2004, 41:801–810.
14 J. F. Jiang, J. P. Shi. Bistability Dynamics in some Structured Ecological Models,in“Spatial Ecology”[M]. CRC Press, 2009.
15 P. A. Stephens, W. J. Sutherland, R. P. Freckleton. What Is the Allee Effect[J].Oikos, 1999, 87:1851–1890.
16 H. L. Smith. Monotone Dynamical System (an Introduction to the Theory of Com-petitive and Cooperative Systems)[M]. AMS., 1995.
17 X. Q. Zhao. Dynamical Systems in Population Biology[M]. Springer, 2003.
18 R. Fisher. The Wave of Advance of Advantageous Genes[J]. Annals of Eugenics,1937, 7:355–369.
19 A. Kolmogorov, I. Petrovsy, N. Piskounov. Study of the Diffusion Equation withGrowth of the Quantity of Matter and its Applications to a Biological Problem[J].Bull. Math, 1937, 1:1–25.
20 J. G. Skellam. Random Dispersal in Theoretical Populations[J]. J. Diff. Equ., 1951,38:196–218.
21 H. Kierstead, L. B. Slobodkin. The Size of Water Masses Containing PlanktonBlooms[J]. J. Mar. Res, 1953, 12:141–147.
22 A. M. Turing. The Chemical Basis of Morphogenesis[J]. Phil. Trans. R. Soc.London Ser. B, 1952, 237:37–72.
23 A. J. Lotka. Elements of Physical Biology[M]. Williams and Wilkins, Baltimore,1925.
24 V. Volterra. Fluctuations in the Abundance of Species[J]. Nature, 1926, 118:558.
25 C. S. Holling. The Components of Predation as Revealed by a Study of SmallMammal Predation of the European Pine Saw?y[J]. Canadian Entomologist., 91:293–320.
26 V. S. Ivlev. Experimental Ecology of the Feeding of Fishes[M]. Yale UniversityPress, 1955.
27 N. Kazarinov, P. van den Driessche. A Model Predator-prey Systems with Func-tional Rresponse[J]. Oecologia, 1978, 39:125–134.
28 P. Turchin. Complex Population Dynamics: A Theoretical/empirical Synthesis[M].Princeton University Press, 2003.
29 K. S. Cheng. Uniqueness of a Limit Cycle for a Predator-prey System[J]. SIAM J.Math. Anal., 1981, 12:541–548.
30 S. B. Hsu. On Global Stability of a Predator-prey System[J]. Math. Biosci., 1978,39:1–10.
31 S. B. Hsu, J. P. Shi. Relaxation Oscillator Profile of Limit Cycle in Predator-preySystem[J]. Discrete Contin. Dynam. Systems. Ser. B, 2009, 11:893–931.
32 Y. Kuang, H. I. Freedman. Uniqueness of Limit Cycles in Gause-type Models ofPredator-prey Systems[J]. Math. Biosci., 1988, 88:67–84.
33 R. M. May. Limit Cycles in Predator-prey Communities[J]. Science, 1972,177:900–902.
34 M. L. Rosenzweig. Paradox of Enrichment: Destabilization of Exploitation Ecosys-tems in Ecological Time[J]. Science, 1971, 171:385–387.
35 D. Xiao, Z. Zhang. On the Uniqueness and Nonexistence of Limit Cycles forPredator-prey System[J]. Nonlinearity, 2003, 16:1–17.
36 M. L. Rosenzweig, R. MacArthur. Graphical Representation and Stability Condi-tions of Predator-prey Interactions[J]. Amer. Natur., 1963, 97:209–223.
37 F. Albrecht, H. Gatzke, A. Haddad, et al. The Dynamics of Two Interacting Popu-lations[J]. J. Math. Anal. Appl., 1974, 46:658–670.
38 F. Albrecht, H. Gatzke, N. Wax. Stable Limit Cycles in Prey-predator Popula-tions[J]. Science, 1973, 181:1074–1075.
39 G. A. K. van Voorn, L. Hemerik, M. P. Boer, et al. Heteroclinic Orbits IndicateOverexploitation in Predator Prey Systems with a Strong Allee Effect[J]. Math.Biosci., 2007, 209:451–469.
40 N. D. Alikakos. An Application of the Invarianceprinciple to Reaction-diffusionEquations[J]. J. Diff. Equ., 1979, 33:201–225.
41 E. D. Conway, D. Hoff, J. A. Smoller. Large Time Behavior of Solutions of Systemsof Nonlinear Reaction-diffusion Equations[J]. SIAM J. Appl. Math., 1978, 35:1–16.
42 R. S. Cantrell, C. Cosner. Spatial Ecology via Reaction-diffusion Equations, WileySeries in Mathematical and Computational Biology[M]. John Wiley & Sons, Ltd.,Chichester, 2003.
43 C. V. Pao. Nonlinear Parabolic and Elliptic Equations[M]. Plenum Press, NewYork, 1992.
44 J. Smoller. Shock Waves and Reaction-diffusion Equations[M]. Springer-Verlag,New York, 1994.
45叶其孝.反应扩散方程[M].科学出版社, 1994.
46 K. N. Chueh, C. C. Conley, J. A. Smoller. Positively Invariant Regions for Systemsof Nonlinear Diffusion Equations[J]. Indiana Univ. Math., 1983, 17:217–234.
47 N. D. Alikakos. A Lyapunov Functional for a Class of Reaction-diffusion Systems.Modeling and Differential Equations in Biology. Lecture Notes in Pure and Appl.Math[J]. 1980:153–170.
48 P. Demotoni, F. Rothe. Convergence to Homogeneous Equilibrium State for Gen-eralized Volterra-lotka Systems[J]. SIAM J. Appl. Math., 1979, 37:648–663.
49王明新.非线性椭圆方程[M].科学出版社, 2008.
50 F. Q. Yi, J. J. Wei, J. P. Shi. Bifucation Analysis of a Diffusive Predator-prey Systemwith Holling Type-ii Function Response[J]. J. Diff. Equ., 2009, 246:1944–1977.
51 E. D. Conway, D. Hoff, J. A. Smoller. Stability and Bifurcation of Steady StateSolutions for Predator Prey Equations[J]. Adv. in Appl. Math., 1982, 3:288–334.
52 C. V. Pao. On Nonlinear Reaction-diffusion Systems[J]. J. Math. Anal. Appl., 1982,87:165–198.
53 P. Koman, A. W. Leung. A General Monotone Scheme for Elliptic Systems withApplications to Ecological Models[J]. Proc. Roy. Soc. Edin. A, 1986, 102:315–325.
54 J. Blat, K. J. Brown. Bifurcation of Steady State Solutions on Predator Prey andCompetition Systems[J]. Proc. Roy. Soc. Edin. A, 1984, 97:21–34.
55 J. Blat, K. J. Brown. Global Bifurcation of Positive Solutions in some Systems ofElliptic Equations[J]. SIAM. J. Math. Anal., 1986, 17:1339–1353.
56 E. N. Dancer. On Positive Solutions of some Paris of Differential Equations, I[J].Trans. Amer. Math. Soc., 1984, 284:729–743.
57 E. N. Dancer. On Positive Solutions of some Paris of Differential Equations, Ii[J].J. Diff. Equ., 1985, 60:236–258.
58 L. Li. Coexistence Theorem of Steady State for Predator Prey Systems[J]. Trans.Amer. Math. Soc., 1988, 305:143–166.
59 L. Li. On the Uniqueness and Ordering of Steady State of Predator Prey Systems[J].Proc. Roy. Soc. Edin. A, 1988, 110:295–303.
60 E. N. Dancer. On the Existence and Uniqueness of Positive Solutions for CompetingSpecies Models with Diffusion[J]. Tran. Amer. Math. Soc., 1991, 326:829–859.
61 E. N. Dancer. On Uniqueness and Stability for Solutions of Singularly PerturbedPredator Prey Type Equations with Diffusion[J]. J. Diff. Equ., 1993, 102:1–32.
62 Y. Lou, W. M. Ni. Diffusion vs Cross-diffusion: An Elliptic Approach[J]. J. Diff.Equ., 1999, 154:157–190.
63 Y. H. Du, Y. Lou. Some Uniqueness and Exact Multiplicity Results for a PredatorPrey Model[J]. Proc. Roy. Soc. Edin. A, 2001, 131:321–349.
64 Y. H. Du, Y. Lou. Qualitive Behavior of Positive Solutions of a Predator PreyModel: Effects of Saturation[J]. Tran. Amer. Math, Soc., 1997, 349:2443–2475.
65 R. Peng, M. X. Wang. Positive Steady States of the Holling-tanner Predator PreyModel with Diffusion[J]. Proc. Roy. Soc. Edin. A, 2005, 135:149–164.
66 H. Matano. Asymptotic Behavior and Stability of Solutions of Semilinear DiffusionEquations[J]. Publ. Res. Inst. Math. Sci., 1979, 15:401–454.
67 H. Matano, M. Mimura. Pattern Formations in Competition-diffusion Systems inNonconvex Domains[J]. Publ. Res. Inst. Math. Sci., 1983, 19:1049–1079.
68 J. D. Murray. Nonexistence of Wave Solutions for the Class of Reaction-diffusionEquations Given by the Volterra Interaction Population Equations with Diffu-sion[J]. J. Theor. Biol., 1975, 52:259–469.
69 L. A. Segel, J. L. Jackson. Dissipative Structure: An Explaination and an EcologicalExample[J]. J. Theor. Biol., 1975, 37:545–559.
70 L. A. Segel, S. A. Levin. Application of Nonlinear Stability Theory to the Study ofthe Effects of Diffusion on Predator-prey Interactions[J]. AIP Conf. Proc., Amer.Inst. Phys., 1976, 27:123–156.
71 V. Volterra. Lecons Sur La Th?′? Orie Math?′?matique De La Lutte Pour La Vie[M].Gauthiers-Villars, Paris, 1931.
72 L. S. Dai. Nonconstant Periodic Solutions in Predator-prey Systems with Continu-ous Time Delay[J]. Math. Biosci., 1981, 53:149–157.
73 T. Faria. Stability and Bifurcation for a Delayed Predator Prey Model and the Effectof Diffusion[J]. J. Math. Anal. Appl, 2001, 254:433–463.
74 Y. Kuang. Delay Differential Equations with Applications in Population Dynam-ics[M]. Academic Press, New York, 1993.
75 J. J. Wei, M. Y. Li. Global Existence of Periodic Solutions in Tri-neuron NetworkModel with Delays[J]. Phy. D, 2004, 198:106–119.
76 J. J. Wei, S. Ruan. Stability and Bifurcation in a Neural Network Model with TwoDelays[J]. Phy. D, 1999, 130:225–272.
77 J. H. Wu. Theory and Application of Partial Functional Differential Equaitons[M].Spring-Verlag, NewYork, 1989.
78 J. H. Wu. Symmetric Functional Differential Equations and Neural Networks withMemory[J]. Trans. Amer. Math. Soc., 1998, 350:4799–4838.
79 X. Lin, J. W. H. So, J. H. Wu. Center Manifolds for Partial Differential Equationswith Delays[J]. Proc. Roy. Soc. Edin. A, 1992, 122:237–254.
80 T. Faria. Normal Forms and Hopf Bifurcation for Partial Differential Equationswith Delays[J]. Trans. Amer. Math. Soc., 2000, 352:2217–2238.
81 T. Faria, L. T. Magalhaes. Normal Form for Retarded Functional Equations withParameters and Applications to Hopf Bifurcation[J]. J. Diff. Equ., 1995, 122:181–200.
82 T. Faria, L. T. Magalhaes. Normal Form for Retarded Functional Equations andApplications to Bogdanov-takens Singularity[J]. J. Diff. Equ., 1995, 122:201–224.
83 Y. Su, A. Y. Wan, J. J. Wei. Bifurcation Analysis in a Diffusive Food-limite Modelwith Time Delay[J]. Appl. Anal., 2010, 89:1161–1181.
84 Y. Su, J. J. Wei, J. P. Shi. Hopf Bifurcations in a Reaction-diffusion PopulationModel with Delay Effect[J]. J. Diff. Equ., 2009, 247:1156–1184.
85 X. P. Yan, W. T. Li. Stability and Hopf Bifurcation for a Delayed CooperativeSystem with Diffusion Effects[J]. Int. J. Bifur. Chaos., 2008, 18:441–453.
86 M. A. Lewis, P. Kareiva. Allee Dynamics and the Spread of Invading Organisms[J].Theor. Popul. Biol., 1993, 43:141–158.
87 A. Morozov, S. Petrovskii, B. L. Li. Bifurcations and Chaos in a Predator-preySystem with the Allee Effect[J]. Proc. Roy. Soc. London Series B-Biol. Sci., 2004,271:1407–1414.
88 A. Morozov, S. Petrovskii, B. L. Li. Spatiotemporal Complexity of Patchy Invasionin a Predator-prey System with the Allee Effect[J]. J. Theor. Biol., 2006, 238:18–35.
89 S. D. Boukal, W. M. Sabelis, L. Berec. How Predator Functional Responses andAllee Effects in Prey Effect the Paradox of Enrichment and Population Collapses[J].Theor. Popul. Biol., 2007, 72:136–147.
90 H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology andEpidemiology. Theory, Models and Simulation. Chapman Hall/crc Mathematicaland Computational Biology Series[M]. Chapman Hall/CRC, Boca Raton, FL, 2008.
91 S. Petrovskii, A. Morozov, B. L. Li. Regimes of Biological Invasion in a Predator-prey System with the Allee Effect[J]. Bull. Math. Biol., 2005, 67:637–661.
92 A. D. Bazykin. Nonlinear Dynamics of Interacting Populations. World ScientificSeries on Nonlinear Science. Series A: Monographs and Treatises, 11[M]. WorldScientific Publishing, 1998.
93 A. Pablo, G. Eduardo, S. Eduardo. Three Limit Cycles in a Leslie-gower Predator-prey Model with Additive Allee Effect[J]. SIAM J. Appl. Math., 2009, 69:1244–1262.
94 E. Gonza′lez-Olivares, B. Gonza′lez-Yavz, E. Sa′ez, et al. On the Number of LimitCycles in a Predator Prey Model with Non-monotonic Functional Response[J]. Dis-crete Contin. Dynam. Systems. Ser. B, 2006, 6:525–534.
95 S. Ruan, D. Xiao. Global Analysis in a Predator-prey System with NonmonotonicFunctional Response[J]. SIAM J. Appl. Math., 2000, 61:1445–1472.
96 R. Arditi, L. R. Ginzburg. Coupling in Predator-prey Dynamics: Ratio-dependence[J]. J. Theor. Biol., 1989, 139:311–326.
97 S. B. Hsu, T. W. Hwang, Y. Kuang. Global Analysis of the Michaelis-menten-typeRatio-dependent Predator-prey System[J]. J. Math. Biol., 2001, 42:489–506.
98 Y. Kuang, E. Beretta. Global Qualitative Analysis of a Ratio-dependent Predator-prey System[J]. J. Math. Biol., 1998, 36:389–406.
99 D. Xiao, S. Ruan. Global Gynamics of a Ratio-dependent Predator-prey System[J].J. Math. Biol., 2001, 43:268–290.
100 S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M].Springer-Verlag, New York, 1990.
101 S. B. Hsu. Ordinary Differential Equations with Applications[M]. Series on Ap-plied Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,2006.
102 E. D. Conway, J. A. Smoller. Global Analysis of a System of Predator-prey Equa-tions[J]. SIAM J. Appl. Math., 1986, 46:630–642.
103 B. D. Hassard, N. D. Kazarinoff, Y. Wan. Theory and Applications of Hopf Bi-furcation[J]. London Math. Soc. Lecture Note Ser. Cambridge University Press,Cambridge, MA, 1981, 41:539–547.
104 Y. A. Kuznetsov. Elements of Applied Bifurcation Theory[M]. Springer-Verlag,New York, 2004.
105 M. L. Rosenzweig. Why the Prey Curve Has a Hump[J]. Amer. Natur., 1969,103:81–87.
106 D. Xiao, Z. Zhang. On the Existence and Uniqueness of Limit Cycles for General-ized Li?′?nard Systems[J]. J. Math. Anal. Appl., 2008, 343:299–309.
107 X. Zeng, Z. Zhang, S. Gao. On the Uniqueness of the Limit Cycle of the General-ized Lie′nard Equation[J]. Bull. London Math. Soc., 1994, 26:213–247.
108 Z. F. Zhang. Proof of the Uniqueness Theorem of Limit Cycles of GeneralizedLie′nard Equations[J]. Appl. Anal., 1986, 23:63–76.
109 J. Hofbauer, J. So. Multiple Limit Cycles for Predator-prey Models[J]. Math.Biosci., 1990, 99:71–75.
110 Y. Kuang. Global Stability of Gause-type Predator-prey System[J]. J. Math. Biol.,1990, 28:462–474.
111 Y. P. Liu. Geometric Criteria for the Nonexistence of Cycles in Gause-typePredator-prey Systems[J]. Proc. Amer. Math. Soc., 2005, 133:3619–3626.
112 J. C. Alexander, J. A. Yorke. Global Bifurcations of Periodic Orbits[J]. Amer. J.Math., 1978, 100:263–292.
113 S. N. Chow, J. Mallet-Paret. The Fuller Index and Global Hopf Bifurcation[J]. J.Diff. Equ., 1978, 29:66–85.
114 J. A. Polking, D. Arnold. Ordinary Differential Equations Using Matlab. 3rd Edi-tion[M]. Prentice Hall, 2003.
115 A. Dhooge, W. Govaerts, Y. A. Kuznetsov. Matcont: A Matlab Package for Numer-ical Bifurcation Analysis of Odes[J]. ACM Trans. Math. Software, 2003, 29:141–164.
116 H. R. Thieme, T. Dhirasakdanon, Z. Han, et al. Species Decline and Extinction:Synergy of Infectious Disease and Allee Effect[J]. J. Biol. Dyna., 2009, 3:305–323.
117 F. M. Hilker, M. Langlais, H. Malchow. The Allee Effect and Infectious Diseases:Extinction, Multistability, and the (dis-) Appearance of Oscillations[J]. Amer.Natur., 2009, 173:72–88.
118 S. D. Boukal, L. Berec. Single-species Models of the Allee Effect: ExtinctionBoundaries, Sex Ratios and Mate Encounters[J]. J. Theor. Biol., 2002, 218:375–394.
119 Y. Gruntfest, R. Arditi, Y. Dombrovsky. A Fragmented Population in a VaryingEnvironment[J]. J. Theor. Biol., 1997, 185:539–547.
120 E. O. Wilson, W. H. Bossert. A Primer of Population Biology[M]. MA, U.S.A.,1971.
121 Y. Takeuchi. Global Dynamical Properties of Lotka-volterra Systems[M]. WorldScientific Publishing Company, 1996.
122 J. Jacobs. Cooperation, Optimal Density and Low Density Thresholds: Yet AnotherModification of the Logistic Model[J]. Oecologia, 1984, 64:395–398.
123 S. R. Zhou, Y. F. Liu, G. Wang. The Stability of Predator-prey Systems Subject tothe Allee Effects[J]. Theor. Popul. Biol., 2005, 67:23–31.
124 R. M. Nisbet, W. S. C. Gurney. Modelling Fluctuating Populations[M]. Chichester.,1982.
125 Y. H. Du, J. P. Shi. Some Recent Results on Diffusive Predator-prey Models inSpatially Heterogeneous Environment[J]. Fields Inst. Commun. Amer. Math. Soc.,Providence, RI, 2006, 48:141–164.
126 Y. H. Du, J. P. Shi. Allee Effect and Bistability in a Spatially HeterogeneousPredator-prey Model[J]. Trans. Amer. Math. Soc., 2007, 359:4557–4593.
127 J. H. Huang, G. Lu, S. G. Ruan. Existence of Traveling Wave Solutions in a Diffu-sive Predator-prey Model[J]. J. Math. Biol., 2003, 46:132–152.
128 W. Ko, K. Ryu. Qualitative Analysis of a Predator-prey Model with Holling Type IiFunctional Response Incorporating a Prey Refuge[J]. J. Diff. Equ., 2006, 231:534–550.
129 Y. Lou, W. M. Ni. Diffusion, Self-diffusion and Cross-diffusion[J]. J. Diff. Equ.,1996, 131:79–131.
130 X. Y. Chen, P. Pola′cˇik. Gradient-like Structure and Morse Decompositions forTime-periodic One-dimensional Parabolic Equations[J]. J. Dynam. Diff. Equ.,1995, 7:73–107.
131 J. F. Jiang, J. P. Shi. Dynamics of a Reaction-diffusion System of AutocatalyticChemical Reaction[J]. Discrete Contin. Dynam. System., 2008, 21:254–258.
132 K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically Autonomous Semi?ows:Chain Recurrence and Lyapunov Functions[J]. Trans. Amer. Math. Soc., 1995,347:1669–1685.
133 R. S. Cantrell, C. Cosner, V. Hutson. Permanence in Ecological Systems with Spa-tial Heterogeneity[J]. Proc. Roy. Soc. Edin. A, 1993, 123:533–559.
134 S. L. Hollis, R. H. Martin, M. Pierre. Global Existence and Boundedness inReaction-diffusion Systems[J]. SIAM J. Math. Anal., 1987, 18:744–761.
135 D. Henry. Geometric Theory of Eemilinear Parabolic Equations, Lecture Notes inMathematics[M]. Springer-Verlag, Berlin-New York, 1981.
136 R. Peng, J. P. Shi, M. X. Wang. On Stationary Patterns of a Reaction-diffusionModel with Autocalysis and Saturation Law[J]. Nonlinearity, 2008, 21:1471–1488.
137 J. P. Shi. Solution Set of Semilinear Elliptic Equations, Global Bifurcation andExact Multiplicity[M]. World Scientific Publishing, 2010.
138 P. H. Rabinowitz. Minimax Methods in Critical Point Theorey with Applicationsto Differential Equaitons[M]. Conference Board of the Mathematical Sciences Re-gional Conference Series in Mathematics., 1984.
139 P. H. Rabinowitz. Some Global Results for Nonlinear Eigenvalue Problems[J]. J.Funct. Anal., 1971, 7:487–513.
140 M. G. Crandall, P. H. Rabinowitz. Bifurcation from Simple Eigenvalues[J]. J.Funct. Anal., 1971, 8:321–340.
141 J. P. Shi. Semilinear Neumann Boundary Value Problems on a Rectangle[J]. Trans.Amer. Math. Soc., 2002, 354:3117–3154.
142 J. Smoller, A. Wasserman. Global Bifurcation of Steady-state Solutions[J]. J. Diff.Equ., 1981, 39(2):269–290.
143 J. F. Jiang, X. Liang, X. Q. Zhao. Saddle-point Behavior for Monotone Semi?owsand Reaction-diffusion Models[J]. J. Diff. Equ., 2004, 203:313–330.
144 J. Y. Jin, J. P. Shi, J. J. Wei, et al. Patterned Solutions in Diffusive Lengyel-epsteinSystem of Cima Chemical Reaction[J]. Rocky Mountain Jour. Math, 2011, ToAppear.
145 Z. Liu, R. Yuan. The Effect of Diffusion for a Predator Prey System with Non-monotonic Functional Response[J]. Int. J. Bifur. Chaos., 2004, 12:4309–4316.
146 K. Cooke, Z. Grossman. Discrete Delay, Distributed Delay and StabilitySwitches[J]. J. Math. Anal. Appl., 1982, 86:592–672.
147 J. K. Hale. Theory of Functional Differential Equations[M]. Springer, New York,1977.
148 S. N. Chow, J. K. Hale. Method of Bifurcation Theory[M]. Springer-Verlag, NewYork, 1982.
149 E. Beretta, Y. Kuang. Geometric Stability Switch Criteria in Delay Differential Sys-tem with Delay Dependent Parameters[J]. SIAM. J. Math. Anal., 2003, 33:1144–1165.
150 Y. L. Song, J. J. Wei. Local Hopf Bifurcation and Global Periodic Solutions in aDelayed Predator Prey System[J]. J. Math. Anal. Appl., 2005, 301:1–21.
151 Y. H. Du, J. P. Shi. A Diffusive Predator-prey Model with a Protection Zone[J]. J.Diff. Equ., 2006, 229:63–91.
152 R. Peng, J. P. Shi, M. X. Wang. Stationary Pattern of a Ratio-dependent Food ChainModel with Diffusion[J]. SIAM J. Appl. Math., 2007, 65:1479–1503.
153 M. X. Wang. Non-constant Positive Steady States of the Sel’kov Model[J]. J. Diff.Equ., 2003, 190:600–620.