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Effect of Spatial Dispersion on Evolutionary Stability: A Two-Phenotype and Two-Patch Model.

Li Q, Zhang J, Zhang B, Cressman R, Tao Y - PLoS ONE (2015)

Bottom Line: In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics.Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable.This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing, China.

ABSTRACT
In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics. The migration rates from one patch to another are considered to be patch-dependent but independent of individual's phenotype. Our main goal is to reveal the dynamical properties of the evolutionary game in a heterogeneous patchy environment. By analyzing the equilibria and their stabilities, we find that the dynamical behavior of the evolutionary game dynamics could be very complicated. Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable. This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.

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Related in: MedlinePlus

Equilibria of Eq (2) on the p − q plane and their stabilities when c1 > 0 and c2 > 0.The red curves correspond to Eq (10) and the green curves Eq (9). The intersections denoted by black spot correspond to stable equilibria of dynamics (2), and the intersections denoted by black circle correspond to unstable equilibria. Parameters are taken as: In panel a, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.25, α2 = 1.8, c1 = 0.25 and c2 = 0.8. In panel b, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.125, α2 = 0.1, c1 = 0.125 and c2 = 0.8. In panel c, A = [1, 0; 0, 1], B = [5, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel d, A = [0, 1; 1, 0], B = [5, 0; 0, 5] β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel e, A = [0, 5; 1, 0], B = [1, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel f, A = [0, 5; 1, 0], B = [0, 1; 5, 0], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel g, A = [1, 0; 0, 5], B = [5, 0; 0, 1], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel h, A = [1, 0; 0, 1], B = [2, 0; 0, 1], β1 = 0.05, β2 = 0.01, α1 = 1.75, α2 = 2, c1 = 0.25 and c2 = 0.01. The positions of p* and q* are marked by dashed lines (0 < p*, q* < 1, and note that p* and q* may not be ESS).
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pone.0142929.g003: Equilibria of Eq (2) on the p − q plane and their stabilities when c1 > 0 and c2 > 0.The red curves correspond to Eq (10) and the green curves Eq (9). The intersections denoted by black spot correspond to stable equilibria of dynamics (2), and the intersections denoted by black circle correspond to unstable equilibria. Parameters are taken as: In panel a, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.25, α2 = 1.8, c1 = 0.25 and c2 = 0.8. In panel b, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.125, α2 = 0.1, c1 = 0.125 and c2 = 0.8. In panel c, A = [1, 0; 0, 1], B = [5, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel d, A = [0, 1; 1, 0], B = [5, 0; 0, 5] β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel e, A = [0, 5; 1, 0], B = [1, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel f, A = [0, 5; 1, 0], B = [0, 1; 5, 0], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel g, A = [1, 0; 0, 5], B = [5, 0; 0, 1], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel h, A = [1, 0; 0, 1], B = [2, 0; 0, 1], β1 = 0.05, β2 = 0.01, α1 = 1.75, α2 = 2, c1 = 0.25 and c2 = 0.01. The positions of p* and q* are marked by dashed lines (0 < p*, q* < 1, and note that p* and q* may not be ESS).

Mentions: From Eq (9), it is easy to see that for the situation with p* ≠ q*, if the interior equilibrium exists, then it should be in the region (0, p*) × (q*, 1), or (p*, 1) × (0, q*) if Δ1Δ2 > 0, and in the region (0, p*) × (0, q*), or (p*, 1) × (q*, 1) if Δ1Δ2 < 0. Of course, it is very difficult to get the exactly analytic solutions of Eqs (9) and (10) in general. The numerical analysis suggests that ten interior equilibria can exist (see Fig 3H). To show this, some examples are plotted in Fig 3. All of these examples show clearly that the equilibrium structure of dynamics (2) could be very complicated.


Effect of Spatial Dispersion on Evolutionary Stability: A Two-Phenotype and Two-Patch Model.

Li Q, Zhang J, Zhang B, Cressman R, Tao Y - PLoS ONE (2015)

Equilibria of Eq (2) on the p − q plane and their stabilities when c1 > 0 and c2 > 0.The red curves correspond to Eq (10) and the green curves Eq (9). The intersections denoted by black spot correspond to stable equilibria of dynamics (2), and the intersections denoted by black circle correspond to unstable equilibria. Parameters are taken as: In panel a, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.25, α2 = 1.8, c1 = 0.25 and c2 = 0.8. In panel b, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.125, α2 = 0.1, c1 = 0.125 and c2 = 0.8. In panel c, A = [1, 0; 0, 1], B = [5, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel d, A = [0, 1; 1, 0], B = [5, 0; 0, 5] β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel e, A = [0, 5; 1, 0], B = [1, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel f, A = [0, 5; 1, 0], B = [0, 1; 5, 0], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel g, A = [1, 0; 0, 5], B = [5, 0; 0, 1], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel h, A = [1, 0; 0, 1], B = [2, 0; 0, 1], β1 = 0.05, β2 = 0.01, α1 = 1.75, α2 = 2, c1 = 0.25 and c2 = 0.01. The positions of p* and q* are marked by dashed lines (0 < p*, q* < 1, and note that p* and q* may not be ESS).
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pone.0142929.g003: Equilibria of Eq (2) on the p − q plane and their stabilities when c1 > 0 and c2 > 0.The red curves correspond to Eq (10) and the green curves Eq (9). The intersections denoted by black spot correspond to stable equilibria of dynamics (2), and the intersections denoted by black circle correspond to unstable equilibria. Parameters are taken as: In panel a, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.25, α2 = 1.8, c1 = 0.25 and c2 = 0.8. In panel b, A = [0, 1; 1, 0], B = [0, 5; 5, 0], β1 = β2 = 0.01, α1 = 1.125, α2 = 0.1, c1 = 0.125 and c2 = 0.8. In panel c, A = [1, 0; 0, 1], B = [5, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel d, A = [0, 1; 1, 0], B = [5, 0; 0, 5] β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel e, A = [0, 5; 1, 0], B = [1, 0; 0, 5], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel f, A = [0, 5; 1, 0], B = [0, 1; 5, 0], β1 = β2 = 0.001, α1 = 1.5, α2 = 1.2, c1 = 0.5 and c2 = 0.2. In panel g, A = [1, 0; 0, 5], B = [5, 0; 0, 1], β1 = β2 = 0.001, α1 = 1.4, α2 = 1.5, c1 = 0.4 and c2 = 0.5. In panel h, A = [1, 0; 0, 1], B = [2, 0; 0, 1], β1 = 0.05, β2 = 0.01, α1 = 1.75, α2 = 2, c1 = 0.25 and c2 = 0.01. The positions of p* and q* are marked by dashed lines (0 < p*, q* < 1, and note that p* and q* may not be ESS).
Mentions: From Eq (9), it is easy to see that for the situation with p* ≠ q*, if the interior equilibrium exists, then it should be in the region (0, p*) × (q*, 1), or (p*, 1) × (0, q*) if Δ1Δ2 > 0, and in the region (0, p*) × (0, q*), or (p*, 1) × (q*, 1) if Δ1Δ2 < 0. Of course, it is very difficult to get the exactly analytic solutions of Eqs (9) and (10) in general. The numerical analysis suggests that ten interior equilibria can exist (see Fig 3H). To show this, some examples are plotted in Fig 3. All of these examples show clearly that the equilibrium structure of dynamics (2) could be very complicated.

Bottom Line: In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics.Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable.This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing, China.

ABSTRACT
In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics. The migration rates from one patch to another are considered to be patch-dependent but independent of individual's phenotype. Our main goal is to reveal the dynamical properties of the evolutionary game in a heterogeneous patchy environment. By analyzing the equilibria and their stabilities, we find that the dynamical behavior of the evolutionary game dynamics could be very complicated. Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable. This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.

Show MeSH
Related in: MedlinePlus