About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 487810, 10 pages
http://dx.doi.org/10.1155/2013/487810
Research Article

Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

1College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
2School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China

Received 25 December 2012; Revised 28 February 2013; Accepted 9 March 2013

Academic Editor: Lan Xu

Copyright © 2013 Weiming Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

1. Introduction

In 1952, Turing published one paper [1] on the subject called “pattern formation"—one of the central issues in ecology [2], putting forth the Turing hypothesis of diffusion-driven instability. Pattern formation in mathematics refers to the process that, by changing a bifurcation parameter, the spatially homogeneous steady states lose stability to spatially inhomogeneous perturbations, and stable inhomogeneous solutions arise [3]. Turing’s revolutionary idea was that passive diffusion could interact with the chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetry, so that the system with diffusion can have them [4]. From then on, pattern formation has become a very active area of research, motivated in part by the realization that there are many common aspects of patterns formed by diverse physical, chemical, and biological systems and by cellular automata and reaction-diffusion equations [57]. And the appearance and evolution of these patterns have been a focus of recent research activity across several disciplines [815].

Segel and Jackson [16] were the first to call attention to the Turing’s ideas that would be also applicable in population dynamics. At the same time, Gierer and Meinhardt [17] gave a biologically justified formulation of a Turing model and studied its properties by employing numerical simulation. Levin and Segel [18, 19] suggested this scenario of spatial pattern formation as a possible origin of planktonic patchiness.

The understanding of patterns and mechanisms of spatial dispersal of interacting species is an issue of significant current interest in conservation biology, ecology, and biochemical reactions [2022]. The spatial component of ecological interaction has been identified as an important factor in how ecological communities are shaped. Empirical evidence suggests that the spatial scale and structure of environment can influence population interactions [23]. A significant amount of work has been done by using this idea in the field of mathematical biology by Murray [20], Okubo and Levin [21], Cantrell and Cosner [23], and others [3, 2427].

In general, assume that the species prey and predator move randomly on spatial domain, and the spatial movement of the individuals is modeled by diffusion with diffusion coefficients for the prey and predator , respectively. As an example, a prototypical predator-prey interaction model with logistic growth rate of the prey in the absence of predation is of the following form [28, 29]: where and are the densities of the prey and predator at time , respectively. And is the Laplacian operator in two-dimensional space.

In recent years, many studies, for example, [3040] and the references therein, show that the reaction-diffusion predator-prey model (e.g., model (1)) is an appropriate tool for investigating the fundamental mechanism of complex spatiotemporal predation dynamics. Of them, Alonso et al. [30] studied how diffusion affects the stability of predator-prey coexistence equilibria and show a new difference between ratio- and prey-dependent models; that is, the prey-dependent models cannot give rise to spatial structures through diffusion-driven instabilities; however, predator-dependent models with the same degree of complexity can. Baurmann et al. [31] investigated the emergence of spatiotemporal patterns in a generalized predator-prey system, derived the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses discussed their biological implications, identified the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation, and found that these bifurcations give rise to complex pattern formation processes in their neighborhood. And Banerjee and Petrovskii [36] studied possible scenarios of pattern formation in a ratio-dependent predator-prey system and found that the emerging patterns are stationary in the large time limit and exhibit only an insignificant spatial irregularity, and spatiotemporal chaos can indeed be observed but only for parameters well inside the Turing-Hopf parameter domain, away from the bifurcation point. Rodrigues et al. [40] paid their attentions to system properties in a vicinity of the Turing-Hopf bifurcation of the predator-prey and found that the asymptotical stationary pattern arises as a sudden transition between two different patterns.

On the other hand, in the research of population dynamics, Allee effect in the population growth has been studied extensively. Allee effect, named after ecologist Allee [41], is a phenomenon in biology characterized by a positive correlation between population size or density and the mean individual fitness (often times measured as per capita population growth rate) of a population or species [42] and may occur under several mechanisms, such as difficulties in finding mates when population density is low, social dysfunction at small population sizes, and increased predation risk due to failing flocking or schooling behavior [4345]. In an ecological point of view, Allee effect has been modeled into strong and weak cases. The strong Allee effect introduces a population threshold, and the population must surpass this threshold to grow. In contrast, a population with a weak Allee effect does not have a threshold. It has been attracting much more attention recently owing to its strong potential impact on the population dynamics of many plants and animal species [46]. Detailed investigations relating to Allee effect may be found in [4759].

In most predation models, it has been considered that Allee effect influences only the prey population. For instance, in model (1), corresponding to the function of prey growth rate of the prey , to express Allee effect, the most usual continuous growth of the equation that is given as: is called additive Allee effect, which was first deduced in [43] and applied in [6062]. Here, is the term of additive Allee effect and and are Allee-effect constants. If , then , and is called weak Allee effect; if , then , and   is strong Allee effect.

Corresponding to model (1), a prototypical predator-prey interaction model with Allee effect on the prey is given by

According to Turing’s idea [1], for model (1)—the special case of model (3) without Allee effect (i.e., )—if the positive equilibrium point is stable in the case (the nonspatial model) but unstable with respect to solutions in the cases and (the spatial model), then is called diffusion-driven instability (i.e., Turing instability or Turing bifurcation), and model (1) may exhibit Turing pattern formation. In contrast, if is stable in the cases and , then there is nonexistence of diffusion-driven instability for model (1), and the model cannot exhibit any pattern formation. And in this situation, for model (3), with Allee effect on the prey, there comes a question: is there any instability of the positive equilibrium occurring? Or, is there any diffusion-driven instability of the positive equilibrium occurring? In addition, does model (3) exhibit any pattern formation controlled by Allee effect?

The goal of this paper is to make an insight into the instability induced by the Allee effect in model (3). Our main interest is to check whether the Allee effect is a plausible mechanism of developing spatiotemporal pattern in the model.

The paper is organized as follows. In the next section, we give the model and stability of the equilibria. In Section 3, we discuss the stability/instability of the spatial model with/without Allee effect, derive the conditions for the occurrence of Allee-diffusion-driven instability of the case with Allee effect, and illustrate typical Turing patterns via numerical simulations. Finally, conclusions and remarks are presented in Section 4.

2. The Model System

In model (1), the product gives the rate at which prey is consumed. The prey consumed per predator, , was termed as the functional response by Solomon [63]. These functions can be defined in different ways. In this paper, following Lotka [64], we adopt which is a linear functional response without saturation, where denotes the capture rate [65]. And following Harrison [28, 29], we set where represents a reduction in the predation rate at high predator densities due to mutual interference among the predators while searching for food.

The proportionality constant is the rate of prey consumption. And the function is given by where denotes the natural death rate of the predator, and can be used to model predator intraspecific competition that is not the direct competition for food, such as some type of territoriality [28]. In this paper, we will discuss the case , which is used in a much more traditional case.

Based on the previous discussions, we can establish the following predation model of two partial differential equations with additive Allee effect on prey: with the positive initial conditions: and the zero-flux boundary conditions: where denotes conversion rate, and is a bounded open domain in with boundary . is the outward unit normal vector on , and zero-flux conditions reflect the situation where the population cannot move across the boundary of the domain.

The main purpose of this paper is to focus on the impacts of diffusion or/and Allee effect on the model system about the positive equilibrium, especially for the instability and pattern formation.

3. Stability Analysis

3.1. The Case without Allee Effect

We first consider the stability of the positive equilibria of model (7) without Allee effect; that is, , and the model is given by

Easy to know that model (10) has a unique positive equilibrium with , where which is locally asymptotically stable. Next, we will discuss the effect of diffusion on .

Set , and the linearized system (10) around is as follows:

Following Malchow et al. [66], we know that any solution of system (12) can be expanded into a Fourier series as follows: where and , . and , are the corresponding wavenumbers.

Having substituted and into (12), we obtain where .

A general solution of (14) has the form , where the constants and are determined by the initial conditions (8) and the exponents are the eigenvalues of the following matrix:

Correspondingly, arises as the solution of following equation: where the trace and determinant of are, respectively,

It is easy to know that and . Hence, the positive equilibrium of model (10) is uniformly asymptotically stable.

Obviously, there is no effect on the stability of the positive equilibrium whether model (10) with diffusion or not. That is to say, there is nonexistence of diffusion-driven instability in model (10), which is the special case of model (7) without Allee effect.

3.2. The Case with Allee Effect
3.2.1. Allee-Diffusion-Driven Instability

In this subsection, we restrict ourselves to the stability analysis of spatial model (7), which is in the presence of Allee effect on prey.

For the sake of learning the effect of Allee effect on the positive equilibrium of model (7), we first give a definition called Allee-diffusion-driven instability as follows.

Definition 1. If a positive equilibrium is uniformly asymptotically stable in the reaction-diffusion model without Allee-effect (e.g., model (10)) but unstable with respect to solutions of the reaction-diffusion model with Allee effect (e.g., model (7)), then this instability is called Allee-diffusion-driven instability.

Next, we will only investigate the stability of the positive equilibrium of model (7). For simplicity, we take the weak Allee effect case as an example, and the unique positive equilibrium is named . We first give the stability of in the case without diffusion as follows that is, in model (7):

The Jacobian matrix of (18) evaluated in the positive equilibrium takes the form: Suppose that , and set

By some computational analysis, we obtain , . Hence is locally asymptotically stable.

And the Jacobian matrix of model (7) at is given by and the characteristic equation of at is where

And the instability sets in when at least or is violated.

Since , is always true. Hence, only violation of gives rise to Allee-diffusion-driven instability, which leads to otherwise, for all if .

Notice that achieves its minimum at the critical value where And is equivalent to where is equivalent to , which is equivalent to

And from , we can determine and as

In conclusion, if , then , and the positive equilibrium of model (7) is unstable. That’s to say, Allee-diffusion-driven instability occurs, and model (7) may exhibit Turing pattern formation.

3.2.2. Pattern Formation

In this subsection, in two-dimensional space, we perform extensive numerical simulations of the spatially extended model (7) in the case with weak Allee effect and show qualitative results. All of the numerical simulations employ the zero-flux boundary conditions (9) with a system size of . Other parameters are fixed as , , , , , , , , and .

The numerical integration of model (7) is performed by using an explicit Euler method for the time integration [67] with a time step size and the standard five-point approximation [68] for the Laplacian with the zero-flux boundary conditions. The initial conditions are always a small amplitude random perturbation around the positive constant steady state solution . After the initial period during which the perturbation spreads, the model goes into either a time-dependent state or an essentially steady state solution (time-independent state).

In the numerical simulations, different types of dynamics can be observed, and it is found that the distributions of predator and prey are always of the same type. Consequently, we can restrict our analysis of pattern formation to one distribution. We only show the distribution of prey as an instance.

In Figure 1, with , there is a pattern consisting of blue hexagons (minimum density of ) in a red (maximum density of ) background, that is, isolated zones with low population densities. We call this pattern as “holes.”

fig1
Figure 1: Typical Turing patterns of in model (7) with parameters , , , , , , , , , and . Times: (a) 0; (b) 50; (c) 250; (d) 2500.

When increasing to , the model dynamics exhibits a transition from stripes-holes growth to stripes replication; that is, holes decay and the stripes pattern emerges (c.f., Figure 2).

fig2
Figure 2: Typical Turing patterns of in model (7) with parameters , , , , , , , , , and . Times: (a) 0; (b) 50; (c) 250; (d) 2500.

When increasing to , the later random perturbations make these stripes decay, end with the time-independent regular spots (c.f., Figure 3), which is isolated zones with high prey densities.

fig3
Figure 3: Typical Turing patterns of in model (7) with parameters , , , , , , , , , and . Times: (a) 0; (b) 50; (c) 250; (d) 2500.

In Figure 4, we show patterns of time-independent stripes-holes and stripes-spots mixture obtained with model (7). These two patterns are similar to each other. With (c.f., Figure 4(a)), the stripes-holes mixture pattern is at relatively low prey densities, while (c.f., Figure 4(b)), at high prey densities.

fig4
Figure 4: Typical Turing patterns of in model (7) with parameters , , , , , , , , and . (a) ; (b) .

From Figures 1 to 4, one can see that, on increasing the control parameter , the pattern sequence “holes stripes-holes mixture stripes stripes-spots mixture spots" is observed.

From the viewpoint of population dynamics, “spots" pattern (c.f., Figure 3) shows that the prey population is driven by predator to a very low level in those regions. The final result is the formation of patches of high prey density surrounded by areas of low prey densities [30]. That is to say, under the control of these parameters, the prey is predominant in the domain. In contrast, “holes" pattern (c.f., Figure 1) indicates that the predator is predominant in the domain.

4. Conclusions and Remarks

In summary, in this paper, we have investigated the spatiotemporal dynamics of a predator-prey model that involves Allee effect on prey analytically and numerically.

For model (7), in the case without Allee effect, there is no effect on the stability of the positive equilibrium whether with diffusion or not. That is to say, there is nonexistence of diffusion-driven instability in the model without Allee effect. More precisely, the distribution of species converge to a spatially homogeneous steady state which varies in time.

And in the case with Allee effect, the positive equilibrium may be unstable. This instability is induced by Allee effect and diffusion together, so we give a new definition called “Allee-diffusion-driven instability" and present the analysis of this instability of the model in details. To the best of our knowledge, this is the first reported case. Furthermore, via numerical simulations, it is found that the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixtures, stripes, stripes-spots mixtures, and spots replication. That is to say, the distribution of species is aggregation. This indicates that the pattern formation of the model with Allee effect is not simple, but rich and complex.

In fact, for a predator-prey system, Okubo and Levin [21] noted Allee effect on the functional response, and a density-dependent death rate of the predator is necessary to generate spatial patterns. And in this paper, we show that a predator-prey system with Allee effect on prey can generate complex Turing spatial patterns, which may be a supplementary to [21].

It is needed to note that, in this paper, we investigate the dynamics of localized patterns in model (7). Such patterns are characterized by a highly spatially heterogeneous solutions and are far from the spatially uniform state. These patterns occur in two-component systems when the ratio of the two diffusion coefficients are very large. In the numerical simulations, we take the diffusivity ratio as , and so we are close to the regime of localized patterns. And the existence of these spatial patterns can be rigorously proved using tools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13, 14], this is desirable in future studies.

Acknowledgments

The authors would like to thank the anonymous referee for very helpful suggestions and comments which led to improvements of our original paper. This research was supported by Natural Science Foundation of Zhejiang Province (LY12A01014 and LQ12A01009) and the National Basic Research Program of China (2012 CB 426510).

References

  1. A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society B, vol. 237, pp. 37–72, 1952.
  2. S. Levin, “The problem of pattern and scale in ecology,” Ecology, vol. 73, no. 6, pp. 1943–1967, 1992.
  3. Z. A. Wang and T. Hillen, “Classical solutions and pattern formation for a volume filling chemotaxis model,” Chaos, vol. 17, no. 3, Article ID 037108, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. N. F. Britton, Essential Mathematical Biology, Springer, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. C. Cross and P. H. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of Modern Physics, vol. 65, no. 3, pp. 851–1112, 1993.
  6. K. J. Lee, W. D. McCormick, Q. Ouyang, and H. L. Swinney, “Pattern formation by interacting chemical fronts,” Science, vol. 261, no. 5118, pp. 192–194, 1993.
  7. A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow, and B.-L. Li, “Spatiotemporal complexity of plankton and fish dynamics,” SIAM Review, vol. 44, no. 3, pp. 311–370, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W.-M. Ni and M. Tang, “Turing patterns in the Lengyel-Epstein system for the CIMA reaction,” Transactions of the American Mathematical Society, vol. 357, no. 10, pp. 3953–3969, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. B. Hoyle, Pattern Formation: An Introduction to Methods, Cambridge University Press, Cambridge, UK, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. J. Ward and J. Wei, “The existence and stability of asymmetric spike patterns for the Schnakenberg model,” Studies in Applied Mathematics, vol. 109, no. 3, pp. 229–264, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. J. Ward, “Asymptotic methods for reaction-diffusion systems: past and present,” Bulletin of Mathematical Biology, vol. 68, no. 5, pp. 1151–1167, 2006.
  12. J. Wei, “Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability,” Physica D, vol. 148, no. 1-2, pp. 20–48, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Wei and M. Winter, “Stationary multiple spots for reaction-diffusion systems,” Journal of Mathematical Biology, vol. 57, no. 1, pp. 53–89, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. Chen and M. J. Ward, “The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model,” SIAM Journal on Applied Dynamical Systems, vol. 10, no. 2, pp. 582–666, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. McKay and T. Kolokolnikov, “Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension,” Discrete and Continuous Dynamical Systems B, vol. 17, no. 1, pp. 191–220, 2012. View at MathSciNet
  16. L Segel and J. Jackson, “Dissipative structure: an explanation and an ecological example,” Journal of Theoretical Biology, vol. 37, no. 3, pp. 545–559, 1972.
  17. A. Gierer and H. Meinhardt, “A theory of biological pattern formation,” Biological Cybernetics, vol. 12, no. 1, pp. 30–39, 1972.
  18. S. Levin and L. Segel, “Hypothesis for origin of planktonic patchiness,” Nature, vol. 259, no. 5545, p. 659, 1976.
  19. S. A. Levin and L. A. Segel, “Pattern generation in space and aspect,” SIAM Review A, vol. 27, no. 1, pp. 45–67, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. D. Murray, Mathematical Biology, Springer, 2003. View at MathSciNet
  21. A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, NY, USA, 2nd edition, 2001. View at MathSciNet
  22. C. Neuhauser, “Mathematical challenges in spatial ecology,” Notices of the American Mathematical Society, vol. 48, no. 11, pp. 1304–1314, 2001. View at Zentralblatt MATH · View at MathSciNet
  23. R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, West Sussex, UK, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  24. T. K. Callahan and E. Knobloch, “Pattern formation in three-dimensional reaction-diffusion systems,” Physica D, vol. 132, no. 3, pp. 339–362, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Bhattacharyay, “Spirals and targets in reaction-diffusion systems,” Physical Review E, vol. 64, no. 1, Article ID 016113, 2001.
  26. T. Leppänen, M. Karttunen, K. Kaski, and R. Barrio, “Dimensionality effects in Turing pattern formation,” International Journal of Modern Physics B, vol. 17, no. 29, pp. 5541–5553, 2003.
  27. J. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar
  28. G. W. Harrison, “Multiple stable equilibria in a predator-prey system,” Bulletin of Mathematical Biology, vol. 48, no. 2, pp. 137–148, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. G. W. Harrison, “Comparing predator-prey models to Luckinbill’s experiment with didinium and paramecium,” Ecology, vol. 76, no. 2, pp. 357–374, 1995.
  30. D. Alonso, F. Bartumeus, and J. Catalan, “Mutual interference between predators can give rise to Turing spatial patterns,” Ecology, vol. 83, no. 1, pp. 28–34, 2002.
  31. M. Baurmann, T. Gross, and U. Feudel, “Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations,” Journal of Theoretical Biology, vol. 245, no. 2, pp. 220–229, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  32. W. Wang, Q.-X. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E, vol. 75, no. 5, Article ID 051913, p. 9, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  33. W. Wang, L. Zhang, H. Wang, and Z. Li, “Pattern formation of a predator-prey system with Ivlev-type functional response,” Ecological Modelling, vol. 221, no. 2, pp. 131–140, 2010.
  34. R. K. Upadhyay, W. Wang, and N. K. Thakur, “Spatiotemporal dynamics in a spatial plankton system,” Mathematical Modelling of Natural Phenomena, vol. 5, no. 5, pp. 102–122, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. J. Wang, J. Shi, and J. Wei, “Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1276–1304, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. M. Banerjee and S. Petrovskii, “Self-organised spatial patterns and chaos in a ratiodependent predator–prey system,” Theoretical Ecology, vol. 4, no. 1, pp. 37–53, 2011.
  37. M. Banerjee, “Spatial pattern formation in ratio-dependent model: higher-order stability analysis,” Mathematical Medicine and Biology, vol. 28, no. 2, pp. 111–128, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. M. Banerjee and S. Banerjee, “Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model,” Mathematical Biosciences, vol. 236, no. 1, pp. 64–76, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  39. S. Fasani and S. Rinaldi, “Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems,” Ecological Modelling, vol. 222, no. 18, pp. 3449–3452, 2011.
  40. L. A. D. Rodrigues, D. C. Mistro, and S. Petrovskii, “Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space- and time-discrete predator-prey system,” Bulletin of Mathematical Biology, vol. 73, no. 8, pp. 1812–1840, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. W. C. Allee, Animal Aggregations: A Study in General Sociology, AMS Press, 1978.
  42. F. Courchamp, J. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, NY, USA, 2008.
  43. B. Dennis, “Allee effects: population growth, critical density, and the chance of extinction,” Natural Resource Modeling, vol. 3, no. 4, pp. 481–538, 1989. View at Zentralblatt MATH · View at MathSciNet
  44. M. A. McCarthy, “The Allee effect, finding mates and theoretical models,” Ecological Modelling, vol. 103, no. 1, pp. 99–102, 1997.
  45. G. Wang, X. G. Liang, and F. Z. Wang, “The competitive dynamics of populations subject to an Allee effect,” Ecological Modelling, vol. 124, no. 2, pp. 183–192, 1999.
  46. M. A. Burgman, S. Ferson, H. R. Akçakaya, et al., Risk Assessment in Conservation Biology, Chapman & Hall, London, UK, 1993.
  47. M. Lewis and P. Kareiva, “Allee dynamics and the spread of invading organisms,” Theoretical Population Biology, vol. 43, no. 2, pp. 141–158, 1993.
  48. P. A. Stephens and W. J. Sutherland, “Consequences of the Allee effect for behaviour, ecology and conservation,” Trends in Ecology and Evolution, vol. 14, no. 10, pp. 401–405, 1999.
  49. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, “What is the Allee effect?” Oikos, vol. 87, no. 1, pp. 185–190, 1999.
  50. T. H. Keitt, M. A. Lewis, and R. D. Holt, “Allee effects, invasion pinning, and Species’ borders,” American Naturalist, vol. 157, no. 2, pp. 203–216, 2002.
  51. S. R. Zhou, Y. F. Liu, and G. Wang, “The stability of predator–prey systems subject to the Allee effects,” Theoretical Population Biology, vol. 67, no. 1, pp. 23–31, 2005.
  52. S. Petrovskii, A. Morozov, and B.-L. Li, “Regimes of biological invasion in a predator-prey system with the Allee effect,” Bulletin of Mathematical Biology, vol. 67, no. 3, pp. 637–661, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  53. J. Shi and R. Shivaji, “Persistence in reaction diffusion models with weak Allee effect,” Journal of Mathematical Biology, vol. 52, no. 6, pp. 807–829, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. A. Morozov, S. Petrovskii, and B.-L. Li, “Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect,” Journal of Theoretical Biology, vol. 238, no. 1, pp. 18–35, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  55. L. Roques, A. Roques, H. Berestycki, and A. Kretzschmar, “A population facing climate change: joint influences of Allee effects and environmental boundary geometry,” Population Ecology, vol. 50, no. 2, pp. 215–225, 2008.
  56. C. Çelik and O. Duman, “Allee effect in a discrete-time predator-prey system,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1956–1962, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  57. J. Zu and M. Mimura, “The impact of Allee effect on a predator-prey system with Holling type II functional response,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3542–3556, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. J. Wang, J. Shi, and J. Wei, “Predator-prey system with strong Allee effect in prey,” Journal of Mathematical Biology, vol. 62, no. 3, pp. 291–331, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  59. E. González-Olivares, H. Meneses-Alcay, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma, and R. Ramos-Jiliberto, “Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey,” Nonlinear Analysis: Real World Applicationsl, vol. 12, no. 6, pp. 2931–2942, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  60. P. Aguirre, E. González-Olivares, and E. Sáez, “Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1401–1416, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  61. P. Aguirre, E. González-Olivares, and E. Sáez, “Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,” SIAM Journal on Applied Mathematics, vol. 69, no. 5, pp. 1244–1262, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  62. Y. Cai, W. Wang, and J. Wang, “Dynamics of a diffusive predator-prey model with additive Allee effect,” International Journal of Biomathematics, vol. 5, no. 2, Article ID 1250023, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  63. M. E. Solomon, “The natural control of animal populations,” The Journal of Animal Ecology, vol. 18, no. 1, pp. 1–35, 1949.
  64. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925.
  65. C. Jost, Comparing Predator–Prey Models Qualitatively and Quantitatively with Ecological Time-Series Data, Institut National Agronomique, Paris-Grignon, 1998.
  66. H. Malchow, S. V. Petrovskii, and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology, Chapman & Hall, Boca Raton, Fla, USA, 2008. View at MathSciNet
  67. M. R. Garvie, “Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 931–956, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  68. A. Munteanu and R. Solé, “Pattern formation in noisy self-replicating spots,” International Journal of Bifurcation and Chaos, vol. 16, no. 12, pp. 3679–3683, 2006. View at Publisher · View at Google Scholar