Invasion dynamics of epidemic with the Allee effect

Investigating the likely success of epidemic invasion is important in the epidemic management and control. In the present study, the invasion of epidemic is initially introduced to a predator-prey system, both species of which are considered to be subject to the Allee effect. Mathematically, the invasion dynamics is described by three nonlinear diffusion-reaction equations and the spatial implicit and explicit models are designed. By means of extensive numerical simulations, the results of spatial implicit model show that the Allee effect has an opposite impact on the invasion criteria and local dynamics when that on the different species. As the intensity of the Allee effect increases, the domain of epidemic invasion reduces and the system dynamics is changed from the stable state to the limit cycle and finally becomes the chaotic state when the susceptible prey with the Allee effect, but the domain expands and the system dynamics is changed from limit cycle to a table point when the predator is subject to the Allee effect. Results from the spatial explicit model show that the strong intensity of the Allee effect can lead to the catastrophic global extinction of all species in the case of that on the susceptible prey. While the predator with the Allee effect, the increased intensity of which makes spatial species reach a stable state. Furthermore, numerical simulations reveal a certain relationship between the invasion speed and spatial patterns.

[1]  Zhien Ma,et al.  A predator-prey model with infected prey. , 2004, Theoretical population biology.

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

[3]  W. C. Allee.  The social life of animals , 1938 .

[4]  T. Clutton‐Brock,et al.  Population dynamics of obligate cooperators , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[5]  W. O. Kermack,et al.  Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity , 1932 .

[6]  M Gyllenberg,et al.  Allee effects can both conserve and create spatial heterogeneity in population densities. , 1999, Theoretical population biology.

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

[8]  Simon A. Levin,et al.  Spread of invading organisms , 1990, Landscape Ecology.

[9]  Rob Hengeveld,et al.  Dynamics of Biological Invasions , 1989 .

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

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

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

[13]  N. Shigesada,et al.  Biological Invasions: Theory and Practice , 1997 .

[14]  Priyanga Amarasekare,et al.  Spatial dynamics of mutualistic interactions , 2004 .

[15]  Cang Hui,et al.  Habitat destruction and the extinction debt revisited: the Allee effect. , 2009, Mathematical biosciences.

[16]  Cang Hui,et al.  Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System , 2008, Bulletin of mathematical biology.

[17]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[18]  Oktay Duman,et al.  Allee effect in a discrete-time predator-prey system , 2009 .

[19]  P. Amarasekare.  Interactions between local dynamics and dispersal: insights from single species models. , 1998, Theoretical population biology.

[20]  Robert D. Holt,et al.  A Theoretical Framework for Intraguild Predation , 1997, The American Naturalist.

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

[22]  A. Ōkubo,et al.  Di?usion and ecological problems: mathematical models , 1980 .

[23]  Andrew P. Dobson,et al.  Ecology of Infectious Diseases in Natural Populations: Frontmatter , 1995 .

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

[25]  G D Ruxton,et al.  Population dynamic consequences of allee effects. , 2002, Journal of theoretical biology.

[26]  W. O. Kermack,et al.  Contributions to the mathematical theory of epidemics—II. The problem of endemicity , 1991, Bulletin of mathematical biology.

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

[28]  F. Gulland,et al.  Ecology of Infectious Diseases in Natural Populations: Impact of Infectious Diseases on Wild Animal Populations: a Review , 1995 .

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

[30]  Priyanga Amarasekare,et al.  Productivity, dispersal and the coexistence of intraguild predators and prey. , 2006, Journal of theoretical biology.

[31]  H. Malchow.  Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics , 1993, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[32]  R. Holt,et al.  Allee Effects, Invasion Pinning, and Species’ Borders , 2001, The American Naturalist.

[33]  Alan Hastings,et al.  Models of spatial spread: A synthesis , 1996 .

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

[35]  R M May,et al.  The invasion, persistence and spread of infectious diseases within animal and plant communities. , 1986, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

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

[37]  Alan A. Berryman,et al.  Population Systems: A General Introduction , 2008 .

[38]  R. Veit,et al.  Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics , 1994 .

[39]  Ilkka Hanski,et al.  Allee effect and population dynamics in the Glanville fritillary butterfly , 1998 .

[40]  Ezio Venturino,et al.  Epidemics in predator-prey models: disease in the predators. , 2002, IMA journal of mathematics applied in medicine and biology.

[41]  Anders Johansen,et al.  Spatio-temporal self-organization in a model of disease spreading , 1994 .

[42]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

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

[44]  A. D. de Roos,et al.  Impact of Intraguild Predation and Stage Structure on Simple Communities along a Productivity Gradient , 2001, The American Naturalist.

[45]  M. Keeling,et al.  The effects of local spatial structure on epidemiological invasions , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[46]  F. van den Bosch,et al.  Spread of organisms: can travelling and dispersive waves be distinguished? , 2000 .

[47]  Jonathan A. Sherratt,et al.  Periodic travelling waves in cyclic predator–prey systems , 2001 .

[48]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

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