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A New Stochastic Individual-Based Model for Pattern Formation and its Application to Predator–Prey Systems

Journal of Biological Physics volume 34, pages 121–133 (2008)Cite this article

Abstract

Reaction–diffusion theory has played a very important role in the study of pattern formation in biology. However, a group of individuals is described by a single state variable representing population density in reaction–diffusion models, and interaction between individuals can be included only phenomenologically. In this paper, we propose a new scheme that seamlessly combines individual-based models with elements of reaction–diffusion theory and apply it to predator–prey systems as a test of our scheme. In the model, starvation periods and the time to reproductive maturity are modeled for individual predators. Similarly, the life cycle and time to reproductive maturity of an individual prey are modeled. Furthermore, both predators and prey migrate through a two-dimensional space. To include animal migration in the model, we use a relationship between the diffusion and the random numbers generated according to a two-dimensional bivariate normal distribution. Despite the simplicity of this model, our scheme successfully produces logistic patterns and oscillations in the population size of both predator and prey. The peak for the predator population oscillation lags slightly behind the prey peak. The simplicity of this scheme will aid additional study of spatially distributed negative-feedback systems.

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Acknowledgements

We wish to acknowledge the support of the Life-Surveyor Project by MEXT 2005, and the High-Tech Research Center Project for Private Universities. A matching fund subsidy was provided by MEXT, 2005-2007.

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Affiliations

  1. Department of Bioscience and Bioinformatics, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan

    Atsushi Yokoyama, Yoshika Noguchi & Seido Nagano

Authors
  1. Atsushi Yokoyama
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  2. Yoshika Noguchi
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  3. Seido Nagano
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Corresponding author

Correspondence to Seido Nagano.

Appendix

Appendix

1 The Definition of the Local Densities (Occupancies) of Predator and Prey

To introduce the concept of the local density or occupancy of the predator and prey populations, the number of animals within a unit area should fluctuate little across several time steps. Otherwise, the concept of local density loses its meaning. To model animal migration, we have generated random numbers that exhibit a two-dimensional bivariate normal distribution with a variance of σ 2. This means that migrating animals are typically in an area of πσ 2 after a single time step. Thus, in an area of πσ 2, the number of animals fluctuates little across several time steps. We adopt an area of 2σ×2σ rather than πσ 2 as the computer simulation uses a square lattice. Thus, the local densities of predator and prey ( n pred and n prey) are defined as the number of animals occupying an area of 2σ×2σ. As shown in Fig. 10b, the number of gradations increases as the number of cells in a unit area increases, causing patterns to appear more clearly. Usually, predators and prey have different σ values, in which case, different unit areas are used.

Fig. 10
figure 10

a An animal migration during a unit time step. b Local densities (or occupancies) for three different unit areas. A 2σ × 2σ area is chosen instead of π σ2 area because the total space in this simulation is square

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2 Scalability of the Lattice Space

To see the scalability of the lattice space, we compare two cases in Fig. 11. Because the migration distance of an animal is characterized by σ, similar patterns are produced when the distance in real space remains unchanged. But the gradation for the local density or occupancy of the animal increases with σ. Thus, as σ increases, patterns remain similar but appear more clearly.

Fig. 11
figure 11

A comparison of two cases, a σ1 = 1 and b σ2 = 2, to confirm that the lattice space is scalable. Patterns produced in both cases are similar to each other, but the patterns appear more clearly in case b

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3 Relationship between the Lattice Space and Real Space

When we approximate real space with the lattice space, we encounter a problem when animals attempt to move to an already occupied cell. In such a case, we advance the animal’s internal clock without moving it. However, such a problem can be solved, as the size of a unit cell Δ becomes smaller. To explain, consider approximating the real space L x ×L y with the following two kinds of lattice spaces: (a) N x ·Δ a ×N y ·Δ a and (b) (2N x ) · (Δa/2) × (2N y ) · (Δa/2), as shown in Fig. 12, where the unit cell size in lattice (a) is Δ a and in lattice (b) is Δ b  = Δ a /2. In cases (1) and (2), using lattice (a), an animal stays in the same cell or moves to the next cell, respectively. The migration of an animal in real space should be independent of the lattice chosen. However, the probability of finding another animal in the final cell using lattice (b) is one in four, because only one of four cells is the target cell. These cases illustrate that the probability of an animal moving to an occupied cell decreases as Δ decreases. Consequently, our model’s deficiencies gradually disappear, as Δ is decreased, and σ is increased, whereas Δ·σ is kept constant.

Fig. 12
figure 12

The relationship between the lattice space and real space. Where Δ a σ a  = Δ b σ b , Δ b  = Δ a /2, and σ b  = 2σ a . The approximation produced by the model is improved as the size of a cell is decreased. In this study, animals are assumed to be smaller in size than a single cell

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Yokoyama, A., Noguchi, Y. & Nagano, S. A New Stochastic Individual-Based Model for Pattern Formation and its Application to Predator–Prey Systems. J Biol Phys 34, 121–133 (2008). https://doi.org/10.1007/s10867-008-9055-6

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Keywords

  • Pattern formation
  • Stochastic model
  • Cellular automata
  • Reaction–diffusion
  • Predator–prey