The stability of predator-prey systems subject to the Allee effects.

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.

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

[2]  Michael A. McCarthy,et al.  The Allee effect, finding mates and theoretical models , 1997 .

[3]  A. Nicholson,et al.  The Balance of Animal Populations.—Part I. , 1935 .

[4]  V. Volterra.  Variations and Fluctuations of the Number of Individuals in Animal Species living together , 1928 .

[5]  Vito Volterra,et al.  Leçons sur la théorie mathématique de la lutte pour la vie , 1931 .

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

[7]  Laurence D. Mueller,et al.  Stability in Model Populations , 2000 .

[8]  B. Lamont,et al.  Population fragmentation may reduce fertility to zero in Banksia goodii — a demonstration of the Allee effect , 1993, Oecologia.

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

[10]  J. Beddington.  Age distribution and the stability of simple discrete time population models. , 1974, Journal of theoretical biology.

[11]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[12]  M. Gross,et al.  Reduced disease in offspring: A benefit of coloniality in sunfish , 1993, Behavioral Ecology and Sociobiology.

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

[14]  Michael J. Crawley,et al.  Natural Enemies: The Population Biology of Predators, Parasites and Diseases , 1992 .

[15]  P. Garber,et al.  A new interpretation of the social organization and mating system of the Callitrichidae , 1987, International Journal of Primatology.

[16]  Petra Klepac,et al.  Stabilizing dispersal delays in predator-prey metapopulation models. , 2002, Theoretical population biology.

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

[18]  Freckleton.  Biological control as a learning process. , 2000, Trends in ecology & evolution.

[19]  J. Poggiale.  From behavioural to population level: Growth and competition , 1998 .

[20]  Klaus Reinhardt,et al.  Conservation of the red-winged grasshopper, Oedipoda germanica (Latr.): the influence of reproductive behaviour , 2002 .

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

[22]  M. Hassell.  The dynamics of arthropod predator-prey systems. , 1979, Monographs in population biology.

[23]  G. F. Gause.  The struggle for existence , 1971 .

[24]  Scheuring,et al.  Allee Effect Increases the Dynamical Stability of Populations. , 1999, Journal of theoretical biology.

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

[26]  P. H. Leslie.  SOME FURTHER NOTES ON THE USE OF MATRICES IN POPULATION MATHEMATICS , 1948 .

[27]  Gang Wang,et al.  The competitive dynamics of populations subject to an Allee effect , 1999 .