- Research
- Open Access
Pattern self-organization and pattern transition on the route to chaos in a spatiotemporal discrete predator–prey system
- Tousheng Huang^{1},
- Xuebing Cong^{1},
- Huayong Zhang^{1}Email author,
- Shengnan Ma^{1} and
- Ge Pan^{1}
https://doi.org/10.1186/s13662-018-1598-7
© The Author(s) 2018
- Received: 30 October 2017
- Accepted: 11 April 2018
- Published: 9 May 2018
Abstract
A spatiotemporal discrete predator–prey system is investigated for understanding the pattern self-organization on the route to chaos. The discrete system is modelled by a coupled map lattice and shows advection of populations in space. Based on the conditions of stable stationary states and Hopf bifurcation, Turing pattern formation conditions are determined. As the parameter value is changed, self-organization of diverse patterns and complex phase transition among the patterns on the route to chaos are observed in simulations. Ordered patterns of stripes, bands, circles, and various disordered states are revealed. When we zoom in to observe the pattern transition in smaller and smaller parameter ranges, subtle structures for transition process are found: (1) alternation between self-organized structured patterns and disordered states emerges as the main nonlinear characteristic; (2) when the parameter value varies in the level from 10^{−3} to 10^{−4}, a cyclic pattern transition process occurs repeatedly; (3) when the parameter value shifts in the level of 10^{−5} or below, stochastic pattern fluctuation dominates as essential regularity for pattern variations. The results obtained in this research promote comprehending pattern self-organization and pattern transition on the route to chaos in spatiotemporal predator–prey systems.
Keywords
- Self-organization
- Chaos
- Coupled map lattice
- Bifurcation
- Turing instability
- Predator–prey system
1 Introduction
In nature, the patterns are a type of non-uniform macroscopic structures with certain orderliness, and they exist commonly and widespreadly [1, 2], and [3]. After decades of research, pattern dynamics has become an important discipline widely applied in various fields [1, 2], and [4]. In the field of ecology, many researchers found that spatial pattern formation is one of the most basic nonlinear characteristics of ecological systems [5] and [6]. On the one hand, spatial composition of ecological relations is a key factor in determining formation and development of biological communities. For example, in predator–prey systems, the predator exerts effort to capture the prey; in turn, the prey strives to escape from the predator’s hunt. With such predator–prey interactions expanding in space, the dynamics and spatial distribution of predator and prey can be more accurately described [7]. On the other hand, spatial pattern formation presents a widespread existence in ecological systems and has attracted attention of many researchers [8]. For example, in arid and semi-arid areas, the plants are often self-organized into regular spatial patterns due to the competition for water resource. This has been verified by numerous field observations [9].
The investigation on spatial pattern formation in ecological systems extends the approach of temporal models and develops the temporal dynamics and stability to spatiotemporal scale. Since spatial patterns often exist in multi-levels and multi-scales of ecological systems, the pattern self-organization represents an important aspect of ecological complexity, i.e., spatiotemporal complexity [10]. Usually, such spatiotemporal complexity is manifested by pattern diversity, including spotted, striped, labyrinth, and spiral patterns, as well as many spatiotemporally chaotic patterns [3] and [8]. As research on pattern dynamics progressed, the researchers noticed that dynamical destabilization and corresponding pattern formation had similarities with equilibrium phase transition [11]. It was also found that the self-organization of spatial patterns plays a key role in indicating catastrophes in ecological systems [8]. Therefore, the study on spatial pattern formation demonstrates great significance in ecology.
In ecological systems, interactions between species are important features. Among the species interactions, predator–prey relationship is one of the most basic and widespread existing interactions. Since the predator–prey interactions always take place over a range of spatial and temporal scales, nonlinear interactions and spatial heterogeneity can often lead to spontaneous formation of predator–prey patterns [3]. During last four decades, spatial pattern formation in spatiotemporal predator–prey systems has received significant attention from many researchers [3, 12], and [13]. Based on the research of pattern self-organization in predator–prey systems, the comprehension of ecological spatiotemporal complexity is promoted.
Studies in mathematical models are informative in understanding the dynamic relationship between the predator and the prey and their complex properties [3] and [14]. For studying the pattern formation in predator–prey systems, a reaction–diffusion model is the most mainstream theoretical model, of which the nonlinear mechanism for pattern self-organization is known as Turing instability. In 1972, Segel and Jackson first applied reaction–diffusion model to study population dynamics: dissipative instability of the copepods–phytoplankton interactions [15]. In the same year, Gierer and Meinhardt explained the biological mechanism of a reaction–diffusion model and studied the properties of corresponding Turing patterns [16]. In 1976, Levin and Segel published a paper in Nature, suggesting that formation of Turing patterns may be the fundamental mechanism for plankton patchiness [17]. Based upon the classical works of Segel and Jackson [15] and Levin and Segel [16], intense research works have been performed to investigate the self-organization of predator–prey patterns due to diffusive instability, with the application of Turing’s instability theory. The reaction–diffusion models have contributed to revealing and explaining the self-organization of various predator–prey patterns.
Based on the reaction–diffusion model, a new model, coupled map lattice (CML), can be developed [24, 25], and [26]. The CML is a type of spatiotemporally discrete model widely applied in an ecological field. Since the pioneering work of May on the discrete logistic model [14], many studies have revealed that the discrete model can exhibit rich nonlinear dynamics for predator–prey systems [24, 27], and [28]. As is well known, flip bifurcation and Hopf bifurcation are the key for triggering the route to chaos, on which complex dynamical behaviors always emerge, such as periodic window, invariant cycles, and chaotic attractors. The research on the routes to chaos has contributed greatly to better comprehending the ordered and disordered states in predator–prey systems. Moreover, the transition between the ordered and disordered states becomes one of the most important topics and attracts the attention of many researchers [29] and [30]. Via comparison, the researchers have also found that CMLs are more practical in describing nonlinear characteristics and spatiotemporal complexity of predator–prey systems than the continuous reaction–diffusion model. With the application of CMLs, many new attractive results have been produced [12, 13, 18, 24, 25], and [26]. Rodrigues et al. [13] revealed a rich variety of pattern formation scenarios in a space- and time-discrete predator–prey system with strong Allee effect and found spatiotemporal multistability under the effects of different initial conditions. Huang et al. [12] and [18] compared the spatial pattern formation between the reaction–diffusion model and its CML version, demonstrating that the nonlinear mechanisms of CML better capture the dynamical complexity of the predator–prey systems. In particular, CMLs can depict discontinuous properties (e.g., patchy environment or fragmented habitat) of predator–prey systems [25]. By the nonlinear mechanisms of CMLs, the spatiotemporal complexity of predator–prey systems can be further revealed and profoundly understood [12, 25, 31], and [26].
In this research, the CML will be applied to investigate the spatiotemporal complexity of the advection–reaction–diffusion system described by Eqs. (1a)–(1b). In former research, Huang et al. have explored a reaction–diffusion predator–prey system with the same functional response, suggesting that such an investigation with discrete time variable and space variable would discover new nonlinear characteristics and dynamical complexity [12]. Extending the study of Huang et al. [12], this research further investigates two more aspects. First, the influence of advection on the spatiotemporal dynamics of a discrete predator–prey system is still limitedly known and therefore deserves investigation. Second, pattern self-organization and pattern transition on the route to chaos are an important topic which still shows challenge. The exploration in this research is arranged as follows. Section 2 gives the CML model and the basic nonlinear characteristics of the discrete predator–prey system. Section 3 performs the Turing instability analysis and determines the pattern formation conditions. Section 4 demonstrates the numerical simulations, and Sect. 5 provides discussion and conclusions.
2 CML model description and system characteristics
2.1 Description of the CML model
For developing the CML model, a two-dimensional rectangular lattice divided into \(n \times n\) sites by space interval h is considered. Simultaneously, the time is divided into a series of slices with time interval τ. Parameters τ and h are the time scale and the space scale for describing the predator–prey dynamics, respectively. Since the growth, death, feeding, and migration of predator and prey individuals always occur periodically, the dynamics of a predator–prey system can be observed by a particular time scale, which can be defined by the generation span of the predator and prey populations and measures the regeneration time of both populations. On the other hand, the space scale on which spatial movements of predator and prey take place can be defined by maximum size of dwelling sites of predator and prey individuals.
In such a spatiotemporal scale, two discrete state variables are defined as \(N_{ ( i,j,m )}\) and \(P_{ ( i,j,m )}\) (\(i,j \in \{ 1,2,3, \ldots,n \}\)), which describes the prey density and the predator density in \((i, j)\) site at mth iteration (notice that with initial time \(t_{0}\), the time at mth iteration is \(t_{0} + m\tau\)). The prey and predator densities in each site change with time in course of the system dynamics, due to the local inter- and intra-specific interactions as well as migration or dispersal between different sites [25].
2.2 Non-spatial dynamic characteristics of the discrete system
- (1)
\(( N_{0}, P_{0} )\) is unstable regardless of the parameter variations;
- (2)
when \(0 < r\tau< 2\) and \(\frac{\beta\varepsilon K}{B + K} < \eta< \frac{2}{\tau} + \frac{\beta\varepsilon K}{B + K}\), \(( N_{1}, P_{1} )\) is stable;
- (3)\(( N_{2}, P_{2} )\) is stable whenin which$$ a_{11}a_{22} - a_{12}a_{21} < 1,\qquad \vert a_{11} + a_{22} \vert < 1 + a_{11}a_{22} - a_{12}a_{21}, $$(11)$$\begin{aligned}& a_{11} = 1 + r\tau\biggl( 1 - \frac{2N_{2}}{K} \biggr) - \frac{a_{21}}{\varepsilon},\qquad a_{12} = - \frac{\tau\eta^{2}}{\beta \varepsilon^{2}} \biggl( 1 + \frac{B}{N_{2}} \biggr), \\& a_{21} = \frac{r^{2}\tau\varepsilon}{\beta} \biggl( w + \frac {B}{P_{2}} \biggr) \biggl( 1 - \frac{N_{2}}{K} \biggr)^{2},\qquad a_{22} = 1 - \tau\eta- a_{12}\varepsilon. \end{aligned}$$
Each fixed point of map (9) is exactly equivalent to a homogeneous stationary state of the discrete system. Therefore, \(( N_{2}, P_{2} )\) can represent a stable spatially homogeneous stationary state. Moreover, when supercritical Hopf bifurcation occurs, attracting invariant closed curves will emerge in the discrete system, representing the spatially homogeneous oscillating states.
3 Turing instability analysis and pattern formation conditions
Turing instability occurs when spatial symmetry breaking takes place and results in the change from spatially homogeneous states to Turing patterns. According to the previous description, the discrete system has two types of homogeneous states: homogeneous stationary state and homogeneous oscillating state. Turing instability occurring on the homogeneous stationary state is generally called pure Turing instability; Turing instability occurring on the homogeneous oscillating state often comes along with Hopf instability and therefore is called Hopf–Turing instability [12] and [18]. Under the influence of Turing instability, local spatially heterogeneous perturbations on the stable homogeneous states can gradually expand to the global spatial domain.
4 Numerical simulations
Simulations are carried out to exhibit the spatiotemporal dynamics of the discrete system. Based on the calculations in Sect. 3, parametric conditions for numerical simulations can be provided. Combining with the research of Huang et al. [12], the values of the following parameters can be fixed as \(\beta= 0.6\), \(\varepsilon= 1\), \(B = 0.4\), \(w = 0.4\), \(\eta= 0.25\), \(K = 1.8\), and \(r = 0.8\). Simultaneously, we can choose \(C _{1} = 0.1\), \(C _{2} = 0.01\), \(D _{1} = 0.01\), \(D _{2} = 0.1\), \(h = 10\), and \(n = 100\), and shift the value of parameter τ to observe the dynamical variations of the discrete system.
As demonstrated in Fig. 1(b), it can be found that the discrete system enters a chaotic dynamics zone at about \(\tau= 4.2243\) from the maximum Lyapunov exponent larger than one. Fig. 1(c) exhibits the variation of the value of \(\lambda_{m}\), determining the range of τ for occurrence of Turing instability. Explicitly via \(\lambda_{m} = 1\), the threshold value for Turing instability occurrence is at about \(\tau= 3.0714\). This suggests the overlap of a Hopf bifurcation point and a Turing bifurcation point, forming the Hopf–Turing bifurcation point. When the value of parameter τ is larger than the Hopf–Turing bifurcation point, the occurrence of Turing instability along the route to chaos can bring the formation of Turing patterns.
With a large amount of numerical simulations, we find that the discrete system often presents two counter types of heterogeneous states. The first type holds distinguishable self-organized ordered structures in configuration, such as circles, stripes, and lines. This type of heterogeneous states is called ordered patterns in this research. Nevertheless, for another type, the spatial distribution of population density is so irregular, scattered, or chaotic that we can hardly visually recognize any ordered structures in the configuration. Therefore, we named the second type of heterogeneous states disordered states hereinafter.
Moreover, we still choose the parameter τ as the main variable inducing pattern transition on the route to chaos. Mathematically, this parameter shows an equal role to other parameters in the discrete system, which can be also chosen to make similar demonstration. From ecology point of view, parameter τ measures the time scale on which predator–prey dynamics takes place, including the growth, death, predating, feeding, and migration of predator and prey individuals, and can be defined by the generation span of the predator and prey populations. With the change of this parameter, we find that the population regeneration is important for determining the spatiotemporal predator–prey dynamics. In literature, this parameter has been also used to explore the dynamic transition from periodic to chaotic behaviors on the route to chaos [27] and [28]. The values of parameters \(C _{1}\), \(C _{2}\), \(D _{1}\), and \(D _{2}\) are best to range in \((0, 1]\), in which the variation of these parameters may hardly change the trend of pattern transition. Based on the numerical simulations, variation of these parameters mainly controls the configuration and occurrence range of striped patterns.
- (a)Mean value of N pattern (MVN), defined as$$ \mathrm{MVN} = \sum_{i = 1}^{i = n} \sum _{j = 1}^{j = n} N ( i,j,t ) \Big/ n^{2}. $$(26)
- (b)Main states of N pattern (MSN), defined asin which \(\operatorname{pos} ( N ( i,j,t ) )\) represents the occurrence possibility of the state \(N ( i,j,t )\) in the prey pattern. To exhibit the main states of the pattern, we need to remove the states of very low occurrence possibility (such as 0.001). With the given n value as 100, \(\operatorname{pos} ( N ( i,j,t ) ) = 0.001\) means the occurrence frequency of \(N ( i,j,t )\) equals 10. Hence, MSN does not contain the states with occurrence frequency less than 10.$$ \mathrm{MSN} = \bigl\{ N ( i,j,t )| \operatorname{pos} \bigl( N ( i,j,t ) \bigr) \ge0.001 \bigr\} , $$(27)
- (c)Information entropy of N pattern (IEN), defined as$$ \mathrm{IEN} = - \sum\operatorname{pos} \bigl( N ( i,j,t ) \bigr)\log \bigl( \operatorname{pos} \bigl( N ( i,j,t ) \bigr) \bigr). $$(28)
5 Discussion and conclusion
For predator–prey systems, self-organization of ordered patterns, resulting from spatial symmetry breaking induced by Turing instability, also plays a key role in revealing and explaining regular population distribution involved in predation relationship. Previously, research works have been focused upon how the pattern self-organization takes place under Turing instability conditions. A great deal of research works demonstrated that Turing instability can generate diverse and complex patterns in the predator–prey systems [3, 12, 25, 32], and [31]. Particularly, Hopf–Turing instability induces spatial symmetry breaking at homogeneous oscillating states and leads to the formation of oscillatory patterns, where the dynamics of predator and prey is always varying spatially and temporally [12] and [31]. A few research works even found that the Hopf–Turing instability can produce patterns with spatiotemporal chaos, which plays a vital role in the spatiotemporal organization of ecological systems [13] and [35].
- (1)
Hopf bifurcation starts a route to chaos, on which the predator–prey dynamics experiences transition from an invariant closed curve, to complex invariant cycles, and finally to chaotic attractors, with periodic windows repeatedly occurringin-between. The dynamic variation on the route to chaos demonstrates a transition from ordered states to disordered states.
- (2)
Hopf–Turing instability occurring on the route to chaos leads to self-organization of diverse patterns. Ordered patterns of stripes, bands, circles, and various disordered states are revealed. Moreover, tiny variation of parameter value can result in two different patterns, reflecting pattern diversity on the route to chaos.
- (3)
When the information entropy of patterns shows high values, rich self-organized patterns may be indicated.
- (4)
Complex pattern transition takes place on the route to chaos. When we zoom in to observe the pattern transition in smaller and smaller parameter ranges, subtle structures for transition process can be found.
- (5)
Alternation between self-organized structured patterns and disordered patterns emerges as the main nonlinear characteristic for pattern transition. Such alternation reveals that ordered patterns and disordered states can keep in continuous transition from one to the other in the discrete system.
- (6)
When the value of parameter τ varies in the level from 10^{−3} to 10^{−4}, cyclic pattern transition process occurs repeatedly. Such a dynamical phenomenon can be explicitly verified by waved variations of the entropy of patterns.
- (7)
When the value of parameter τ varies in the level of 10^{−5} or below, stochastic pattern fluctuation dominates the pattern transition. Moreover, the pattern fluctuation presents a property of self-similarity, reflecting basic regularity for pattern variations in tiny parameter ranges.
Declarations
Acknowledgements
The authors would like to acknowledge with great gratitude the support of the National Major Science and Technology Program for Water Pollution Control and Treatment (No. 2017ZX07101-002, No. 2015ZX07203-011), the Fundamental Research Funds for the Central Universities (No. JB2017069).
Authors’ contributions
The authors’ contributions are described as follows. HZ provided the innovations for the work, TH and XC wrote and modified the paper together, SM checked the calculations for bifurcation analysis and gave suggestions for paper writing, and GP helped to perform numerical simulations. All authors read and approved the final manuscript.
Competing interests
The authors have no financial and non-financial competing interests for this research work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Cross, M., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press, Cambridge (2009) View ArticleMATHGoogle Scholar
- Nicolis, G., Progogine, I.: Exploring Complexity: An Introduction. Freeman, San Francisco (1989) Google Scholar
- Kondo, S., Miura, T.: Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329, 1616–1620 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Walgraef, D.: Spatio-Temporal Pattern Formation: With Examples from Physics, Chemistry, and Materials Science. Springer, New York (2012) Google Scholar
- Levin, S.A.: Pattern formation in ecological communities. In: Steele, J.S. (ed.) Spatial Pattern in Plankton Communities, pp. 433–465. Plenum, New York (1978) View ArticleGoogle Scholar
- Rietkerk, M., Van de Koppel, J.: Regular pattern formation in real ecosystems. Trends Ecol. Evol. 23(3), 169–175 (2008) View ArticleGoogle Scholar
- Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399(6734), 354–359 (1999) View ArticleGoogle Scholar
- Rietkerk, M., Dekker, S.C., de Ruiter, P.C., van de Koppel, J.: Self-organized patchiness and catastrophic shifts in ecosystems. Science 305(5692), 1926–1929 (2004) View ArticleGoogle Scholar
- Valentin, C., d’Herbès, J.M., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37(1), 1–24 (1999) View ArticleGoogle Scholar
- Bascompte, J., Solé, R.V.: Rethinking complexity: modelling spatiotemporal dynamics in ecology. Trends Ecol. Evol. 10(9), 361–366 (1995) View ArticleGoogle Scholar
- Van den Broeck, C., Parrondo, J.M.R., Toral, R.: Noise-induced nonequilibrium phase transition. Phys. Rev. Lett. 73(25), 3395 (1994) View ArticleGoogle Scholar
- Huang, T., Zhang, H., Yang, H., Wang, N., Zhang, F.: Complex patterns in a space- and time-discrete predator–prey model with Beddington–DeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 43, 182–199 (2017) MathSciNetView ArticleGoogle Scholar
- Rodrigues, L.A.D., Mistro, D.C., Petrovskii, S.: Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect. Theor. Ecol. 5, 341–362 (2012) View ArticleGoogle Scholar
- May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976) View ArticleMATHGoogle Scholar
- Segel, L.A., Jackson, J.L.: Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37(3), 545–559 (1972) View ArticleGoogle Scholar
- Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12(1), 30–39 (1972) View ArticleMATHGoogle Scholar
- Levin, S.A., Segel, L.A.: Hypothesis for origin of planktonic patchiness. Nature 259, 659 (1976) View ArticleGoogle Scholar
- Huang, T., Zhang, H., Yang, H.: Spatiotemporal complexity of a discrete space-time predator–prey system with self- and cross-diffusion. Appl. Math. Model. 47, 637–655 (2017) MathSciNetView ArticleGoogle Scholar
- Wang, C.: Rich dynamics of a predator–prey model with spatial motion. Appl. Math. Comput. 260, 1–9 (2015) MathSciNetGoogle Scholar
- Sun, G.Q., Zhang, J., Song, L.P., Jin, Z., Li, B.L.: Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218, 11151–11162 (2012) MathSciNetMATHGoogle Scholar
- Sun, G.Q., Jin, Z., Liu, Q.X., Li, L.: Dynamical complexity of a spatial predator–prey model with migration. Ecol. Model. 219, 248–255 (2008) View ArticleGoogle Scholar
- Liu, P.P.: An analysis of a predator–prey model with both diffusion and migration. Math. Comput. Model. 51, 1064–1070 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Petrovskii, S., Li, B.L.: An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect. Math. Biosci. 186, 79–91 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Huang, T., Zhang, H.: Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system. Chaos Solitons Fractals 91, 92–107 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Mistro, D.C., Rodrigues, L.A.D., Petrovskii, S.: Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect. Ecol. Complex. 9, 16–32 (2012) View ArticleGoogle Scholar
- Punithan, D., Kim, D.K., McKay, R.I.B.: Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices. Ecol. Complex. 12, 43–57 (2012) View ArticleGoogle Scholar
- Liu, X., Xiao, D.: Complex dynamic behaviors of a discrete-time predator–prey system. Chaos Solitons Fractals 32, 80–94 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Jing, Z., Yang, J.: Bifurcation and chaos in discrete-time predator–prey system. Chaos Solitons Fractals 27, 259–277 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Antal, T., Droz, M.: Phase transitions and oscillations in a lattice prey–predator model. Phys. Rev. E 63, 056119 (2001) View ArticleGoogle Scholar
- Schaffer, W.M.: Order and chaos in ecological systems. Ecology 66, 93–106 (1985) View ArticleGoogle Scholar
- Haque, M.: Existence of complex patterns in the Beddington–DeAngelis predator–prey model. Math. Biosci. 239, 179–190 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, X.C., Sun, G.Q., Jin, Z.: Spatial dynamics in a predator–prey model with Beddington–DeAngelis functional response. Phys. Rev. E 85, 021924 (2012) View ArticleGoogle Scholar
- Bai, L., Zhang, G.: Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions. Appl. Math. Comput. 210, 321–333 (2009) MathSciNetMATHGoogle Scholar
- Han, Y.-T., Han, B., Zhang, L., Xu, L., Li, M.-F., Zhang, G.: Turing instability and wave patterns for a symmetric discrete competitive Lotka–Volterra system. WSEAS Trans. Math. 10, 181–189 (2011) Google Scholar
- Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44, 311–370 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Kaneko, K.: Pattern dynamics in spatiotemporal chaos: pattern selection, diffusion of defect and pattern competition intermittency. Phys. D, Nonlinear Phenom. 34, 1–41 (1989) MathSciNetView ArticleMATHGoogle Scholar