Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

[1]  M. Rosenzweig.  Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time , 1971, Science.

[2]  Kuo-Shung Cheng,et al.  UNIQUENESS OF A LIMIT CYCLE FOR A PREDATOR-PREY SYSTEM* , 1981 .

[3]  Leah Edelstein-Keshet,et al.  Mathematical models in biology , 2005, Classics in applied mathematics.

[4]  Horst Malchow,et al.  Spatiotemporal Complexity of Plankton and Fish Dynamics , 2002, SIAM Rev..

[5]  Sergei Petrovskii,et al.  Allee effect makes possible patchy invasion in a predator-prey system. , 2002 .

[6]  Junjie Wei,et al.  NONEXISTENCE OF PERIODIC ORBITS FOR PREDATOR-PREY SYSTEM WITH STRONG ALLEE EFFECT IN PREY POPULATIONS , 2013 .

[7]  Bernd Krauskopf,et al.  Nonlinear Dynamics of Interacting Populations , 1998 .

[8]  Alan Hastings,et al.  Allee effects in biological invasions , 2005 .

[9]  W. C. Allee.  Animal Aggregations: A Study in General Sociology , 1931 .

[10]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[11]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[12]  Dongmei Xiao,et al.  Global dynamics of a ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[13]  P. Turchin.  Complex Population Dynamics: A Theoretical/Empirical Synthesis , 2013 .

[14]  Yuan Lou,et al.  Diffusion, Self-Diffusion and Cross-Diffusion , 1996 .

[15]  M E Gilpin,et al.  Enriched predator-prey systems: theoretical stability. , 1972, Science.

[16]  S. Hsu,et al.  A Contribution to the Theory of Competing Predators , 1978 .

[17]  Zhang Zhifen,et al.  Proof of the uniqueness theorem of limit cycles of generalized liénard equations , 1986 .

[18]  W. C. Allee.  Animal aggregations, a study in general sociology. / by W. C. Allee. , 1931 .

[19]  Timothy J. Sluckin,et al.  Consequences for predators of rescue and Allee effects on prey , 2003 .

[20]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[21]  Rui Peng,et al.  On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law , 2008 .

[22]  P. Polácik,et al.  Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations , 1995 .

[23]  George A K van Voorn,et al.  Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect. , 2007, Mathematical biosciences.

[24]  Joel Smoller,et al.  Global analysis of a system of predator-prey equations , 1986 .

[25]  Jifa Jiang,et al.  Saddle-point behavior for monotone semiflows and reaction–diffusion models☆ , 2004 .

[26]  J. Alexander,et al.  GLOBAL BIFURCATIONS OF PERIODIC ORBITS. , 1978 .

[27]  N. D. Kazarinoff,et al.  A model predator-prey system with functional response , 1978 .

[28]  H. Wittmer.  Allee Effects in Ecology and Conservation , 2010 .

[29]  Sergei Petrovskii,et al.  Regimes of biological invasion in a predator-prey system with the Allee effect , 2005, Bulletin of mathematical biology.

[30]  Paul Waltman,et al.  Competing Predators , 2007 .

[31]  R M May,et al.  Stable Limit Cycles in Prey-Predator Populations , 1973, Science.

[32]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[33]  Nicola Sottocornola,et al.  Robust homoclinic cycles in Bbb R4 , 2003 .

[34]  H. Bhadeshia.  Diffusion , 1995, Theory of Transformations in Steels.

[35]  M A Lewis,et al.  How predation can slow, stop or reverse a prey invasion , 2001, Bulletin of mathematical biology.

[36]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[37]  Eduardo Sáez,et al.  Three Limit Cycles in a Leslie--Gower Predator-Prey Model with Additive Allee Effect , 2009, SIAM J. Appl. Math..

[38]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[39]  Robert H. Martin,et al.  Global existence and boundedness in reaction-diffusion systems , 1987 .

[40]  Dongmei Xiao,et al.  On the uniqueness and nonexistence of limit cycles for predator?prey systems , 2003 .

[41]  Dongmei Xiao,et al.  On the existence and uniqueness of limit cycles for generalized Liénard systems , 2008 .

[42]  Junjie Wei,et al.  Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of Cima chemical reactions , 2013 .

[43]  I. Szántó,et al.  On the number of limit cycles in a predator prey model with non-monotonic functional response , 2006 .

[44]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[45]  Hal L. Smith,et al.  Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions , 1995 .

[46]  Mingxin Wang,et al.  Non-constant positive steady states of the Sel'kov model ☆ , 2003 .

[47]  Yuan Lou,et al.  DIFFUSION VS CROSS-DIFFUSION : AN ELLIPTIC APPROACH , 1999 .

[48]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[49]  Romuald N. Lipcius,et al.  Allee effects driven by predation , 2004 .

[50]  R. Arditi,et al.  A FRAGMENTED POPULATION IN A VARYING ENVIRONMENT , 1997 .

[51]  Zhang Zhi-fen,et al.  On the Uniqueness of the Limit Cycle of the Generalized Liénard Equation , 1994 .

[52]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[53]  S. Hsu,et al.  Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[54]  Jifa Jiang,et al.  Bistability Dynamics in Structured Ecological Models , 2009 .

[55]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[56]  Ludek Berec,et al.  How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. , 2007, Theoretical population biology.

[57]  C. S. Holling.  The components of prédation as revealed by a study of small-mammal prédation of the European pine sawfly. , 1959 .

[58]  Robert M. May,et al.  Limit Cycles in Predator-Prey Communities , 1972, Science.

[59]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[60]  R. McGehee,et al.  Some mathematical problems concerning the ecological principle of competitive exclusion , 1977 .

[61]  J. Felsenstein.  A primer of population biology , 1972 .

[62]  Ludek Berec,et al.  Multiple Allee effects and population management. , 2007, Trends in ecology & evolution.

[63]  Sergei Petrovskii,et al.  Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. , 2006, Journal of theoretical biology.

[64]  P. Kareiva,et al.  Allee Dynamics and the Spread of Invading Organisms , 1993 .

[65]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[66]  Nicholas D. Alikakos,et al.  An application of the invariance principle to reaction-diffusion equations , 1979 .

[67]  Sze-Bi Hsu,et al.  ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS , 2005 .

[68]  V. Volterra.  Fluctuations in the Abundance of a Species considered Mathematically , 1926, Nature.

[69]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[70]  S. Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology : theory, models, and simulation , 2007 .

[71]  Bai-lian Li,et al.  Bifurcations and chaos in a predator-prey system with the Allee effect , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[72]  Junping Shi,et al.  Persistence in reaction diffusion models with weak allee effect , 2006, Journal of mathematical biology.

[73]  Junjie Wei,et al.  Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey , 2011 .

[74]  Yihong Du,et al.  Allee effect and bistability in a spatially heterogeneous predator-prey model , 2007 .

[75]  Grenfell,et al.  Inverse density dependence and the Allee effect. , 1999, Trends in ecology & evolution.

[76]  Wonlyul Ko,et al.  Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge , 2006 .

[77]  Sze-Bi Hsu,et al.  On Global Stability of a Predator-Prey System , 1978 .

[78]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[79]  V. Volterra.  Fluctuations in the Abundance of a Species considered Mathematically , 1926 .

[80]  Shigui Ruan,et al.  Global Analysis in a Predator-Prey System with Nonmonotonic Functional Response , 2001, SIAM J. Appl. Math..

[81]  Jonathan A. Sherratt,et al.  Oscillations and chaos behind predator–prey invasion: mathematical artifact or ecological reality? , 1997 .

[82]  M. Rosenzweig,et al.  Why the Prey Curve Has a Hump , 1969, The American Naturalist.

[83]  R. Lande,et al.  Extinction Thresholds in Demographic Models of Territorial Populations , 1987, The American Naturalist.

[84]  D. Hoff,et al.  LARGE TIME BEHAVIOR OF SOLUTIONS OF SYSTEMS OF NONLINEAR REACTION-DIFFUSION EQUATIONS* , 1978 .

[85]  John C. Polkinghorne,et al.  Ordinary differential equations using MATLAB , 1999 .

[86]  Gang Wang,et al.  The stability of predator-prey systems subject to the Allee effects. , 2005, Theoretical population biology.

[87]  Michel Langlais,et al.  The Allee Effect and Infectious Diseases: Extinction, Multistability, and the (Dis‐)Appearance of Oscillations , 2008, The American Naturalist.

[88]  Scott Ferson,et al.  Risk assessment in conservation biology , 1993 .

[89]  Y. Takeuchi.  Global Dynamical Properties of Lotka-Volterra Systems , 1996 .

[90]  F. Albrecht,et al.  The dynamics of two interacting populations , 1974 .

[91]  Rui Peng,et al.  Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion , 2007, SIAM J. Appl. Math..

[92]  B. Arms,et al.  Cooperation , 1926, Becoming Rooted.

[93]  Junping Shi,et al.  Bistability dynamics in some structured ecological models ∗ , 2008 .

[94]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[95]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[96]  Yihong Du,et al.  A diffusive predator–prey model with a protection zone☆ , 2006 .

[97]  Jifa Jiang,et al.  DYNAMICS OF A REACTION-DIFFUSION SYSTEM OF AUTOCATALYTIC CHEMICAL REACTION , 2008 .

[98]  Ludek Berec,et al.  Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. , 2002, Journal of theoretical biology.

[99]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[100]  Ralf Blossey,et al.  Computational Biology (Chapman & Hall/Crc Mathematical and Computational Biology Series) , 2006 .

[101]  Thanate Dhirasakdanon,et al.  Species decline and extinction: synergy of infectious disease and Allee effect? , 2009, Journal of biological dynamics.

[102]  P. Haccou.  Mathematical Models of Biology , 2022 .

[103]  Mark Kot,et al.  Elements of Mathematical Ecology , 2001 .

[104]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[105]  Shui-Nee Chow,et al.  The Fuller index and global Hopf bifurcation , 1978 .

[106]  G. Tullock,et al.  Competitive Exclusion. , 1960, Science.

[107]  Yihong Du,et al.  Some Recent Results on Diffusive Predator-prey Models in Spatially Heterogeneous Environment ∗ † , 2005 .

[108]  Sergei Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology , 2007 .

[109]  Sze-Bi Hsu,et al.  Relaxation oscillation profile of limit cycle in predator-prey system , 2009 .

[110]  Leon O. Chua,et al.  WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE , 2009 .

[111]  E. N. Dancer.  MINIMAX METHODS IN CRITICAL POINT THEORY WITH APPLICATIONS TO DIFFERENTIAL EQUATIONS (CBMS Regional Conference Series in Mathematics 65) , 1987 .

[112]  William H. Bossert,et al.  A primer of population biology , 1972 .

[113]  Brian Dennis,et al.  ALLEE EFFECTS: POPULATION GROWTH, CRITICAL DENSITY, AND THE CHANCE OF EXTINCTION , 1989 .

[114]  Joel Smoller,et al.  Global Bifurcation of Steady-State Solutions , 1981 .

[115]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[116]  Junping Shi,et al.  Semilinear Neumann boundary value problems on a rectangle , 2002 .

[117]  M. Kot,et al.  Speeds of invasion in a model with strong or weak Allee effects. , 2001, Mathematical biosciences.

[118]  P. Hartman.  Ordinary Differential Equations , 1965 .

[119]  Junping Shi,et al.  On global bifurcation for quasilinear elliptic systems on bounded domains , 2009 .

[120]  J. Jacobs.  Cooperation, optimal density and low density thresholds: yet another modification of the logistic model , 1984, Oecologia.

[121]  J. Gascoigne,et al.  Allee Effects in Ecology and Conservation , 2008 .

[122]  Sze-Bi Hsu,et al.  Relaxation , 2021, Graduate Studies in Mathematics.

[123]  William J. Sutherland,et al.  What Is the Allee Effect , 1999 .

[124]  Stephens,et al.  Consequences of the Allee effect for behaviour, ecology and conservation. , 1999, Trends in ecology & evolution.

[125]  V. S. Ivlev,et al.  Experimental ecology of the feeding of fishes , 1962 .

[126]  Jianhua Huang,et al.  Existence of traveling wave solutions in a diffusive predator-prey model , 2003, Journal of mathematical biology.

[127]  Yang Kuang,et al.  Uniqueness of limit cycles in Gause-type models of predator-prey systems , 1988 .

[128]  E. F. Infante,et al.  A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type. , 1974 .

[129]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[130]  W. Gurney,et al.  Modelling fluctuating populations , 1982 .