Appendix A: Evolutionary dynamics

Transcription

1 Supporting Information (Appendixes) Appendix A: Evolutionary dynamics The quantitative genetic model of the evolutionary dynamics (Iwasa et al. 1991) is rearranged as follows (Yamauchi and Yamamura 2004): (A1a) (A1b) where T X and T Y are the generation time of each species, G X and G Y are the additive genetic variances of the prey trait and the predator trait, respectively, and the remains on the right-hand side (r.h.s.) are the selection gradient. W X (W Y ) is the fitness of an individual with trait ( ) in a population with mean trait u (v). Eq. (A1) can be rewritten as (A2a) (A2b) where and represent the speed of evolutionary adaptation in the prey and predator, respectively. References Yamauchi, A., and N. Yamamura Effects of defense evolution and diet choice on population dynamics in a one-predator two-prey system. Ecology. 86:

2 Appendix B: Stability analysis of a four-dimensional system We study the local stability of the system described by Eqs. (1) and (2) by linearizing the dynamics near the nontrivial equilibrium. We can judge the local stability by whether the characteristic equation of their Jacobian matrix satisfies the Routh-Hurwitz criteria. We can obtain the Jacobian matrix (not shown) under the equilibrium condition, (B1a) (B1b), (B1c), (B1d) where primes denote first derivatives. In the specific function a(u,v),. Thus, at equilibrium. This result implies that at evolutionary equilibrium, the proportionate cost of increasing the value of the trait of one species must balance with the cost of increasing the value of the trait of the other species. The characteristic equation for determining the eigenvalues is. The equilibrium point is locally stable if and, according to the Routh-Hurwitz criteria. Hence, we have, (B2a)

3 , (B2b), (B2c), (B2d) where and (we set B 1 (< 0) = C 1 (< 0) at equilibrium and used the relations obtained from ; double primes indicate the second derivative). In inequality (B3d), and equal the left-hand sides of (B2a), (B2b), and (B2c), respectively. When (A = 0 because = 0), the criteria (B2a) (B2c) are always satisfied when B 2 < 0 and C 2 < 0. (B2d) is likely to be satisfied if <<1 (because the l.h.s. can be positive if, and the r.h.s. can be negative if << 1 (note that > 1)). In a similar way, we can examine the cases when. When (A < 0 because < 0), the criteria are likely to be satisfied if B 2 << 0, C 2 < 0 and << 1. When (A > 0 because > 0), the criteria are likely to be satisfied if B 2 < 0, C 2 << 0 and << 1. In the special case in which the trade-off functions are linear (B 2, C 2 = 0), the criteria are simplified as follows,, (B3a). (B3b)

4 When, (B3b) is satisfied if. When (A < 0), the criteria are likely to be satisfied if / > r(u * )/d, C 1 <<1, <<1, and <<1. When (A > 0), the criteria are likely to be satisfied if / < r(u * )/d and >>1.

5 Appendix C: Local stability analysis when coevolution is fast We analyze the local stability of equilibrium in the case that the adaptive speed is very fast. This analysis also can be applied to cases in which the trait dynamics describe phenotypic plasticity instead of speed of genetic replacement. When is very large, both species should behave in a way that maximizes their fitness very quickly. Thus, we consider the local stability of equilibrium in a predator prey system (1) for and The conditions and can be rewritten as,, (C1a), (C1b) where a = a(u,v) and g = g(v). From (C1), we obtain. (C2) The equilibrium of system (1) is locally stable if. (C3) From (C2), the stability condition is. (C4) From (C1), we obtain. (C5) Hence, (C4) becomes. (C6) Since only g depends on Y, the stability condition is. (C7) By differentiating (C5) with respect to Y, we can rewrite ineq. (C7) as

6 . (C8) From (C5), we find that the numerator of the r.h.s. of (C8) is positive. Thus, (C8) becomes. (C9) Since at equilibrium, (C9) becomes. (C10) Using the equilibrium, we obtain. (C11) This can be rewritten as ineq. (3) in the text.

7 Appendix D: Local stability analysis when coevolution is slow When the evolutionary dynamics are much slower than the population dynamics ( <<1), the coevolutionary dynamics are approximated by averaged dynamics: (D1a), (D1b) where <> represents the temporal average with respect to the population cycles when u or v is fixed. In this system, the population dynamics shows cycles; thus, we need to use the mean abundance of one cycle instead of the equilibrium abundance. We have (D2a). (D2b) Then (D1) can be rewritten as, (D3a). (D3b) By substituting the equilibrium values of the population dynamics, X * and Y *, while keeping trait values fixed, we obtain the evolutionary dynamics equations as follows,, (D4a). (D4b) First, we study the local stability of this system. Under the equilibrium condition, (D5a)

8 . (D5b) We obtain the Jacobian matrix, (D6) where,,,,,, and. In the following analysis, we use assumptions based on the specific function, where. By using the Routh-Hurwitz criteria, we can evaluate the local stability. If the trace of J is negative and the determinant of J is positive, the system is stable. The local stability condition is, (D7a), (D7b) where the notation is the same as in Appendix B. We note that (D7a) and (D7b) are respectively the same as (B3a) and (B3c). Next, we study the conditions under which the above stability criteria are likely to be satisfied. When (A = 0), ineq. (D7) is always satisfied if B 2 and C 2 < 0. When (A < 0), the criteria are likely to be satisfied if and. When (A > 0), the criteria are likely to be satisfied if and. In the special case in which the trade-off functions are linear (B 2, C 2 = 0), the criteria are simplified as follows,

9 , (D8) because (D7b) is always satisfied. We note that this criterion is the same as (B4a). When u * > v * (A > 0), the system is locally stable if. (D9) When u * < v * (A < 0), the system is always locally stable if the ineq. opposite to (D9) holds. When u * = v * (A = 0), the system is always locally stable.

10 Appendix E: Proof of asymmetric oscillations in the case of slower adaptive speeds Here we show that the amplitude of fluctuations in the prey s trait is much larger than that the amplitude of fluctuations in the predator s trait when evolution is slow (see Figs. 3a, 4a). Consider the situation where the trait values u and v oscillate slightly around the equilibrium trait values. We define u and v as u = u * + u and v = v * + v. Then the dynamics of u and v can be approximated as, (E1a), (E1b) where,,, and. The equations (E1) can be rewritten as, (E2a). (E2b) From (E2), the asymmetric oscillation pattern is clear.

11 Appendix F: Local stability analysis in the case with density-dependent prey growth When the prey s population shows density-dependent growth, the system without adaptation is always globally stable. We can then analyze how adaptation affects the system. Without loss of generality, we can rescale the dynamics and choose K = 1. First, we show the stability criteria in a four-dimensional system for the specific case in which the trade-off functions are linear ( ) and the same in both species ( ). Second, we show the stability criteria of a two-dimensional system in which the evolutionary dynamics is much slower than the population dynamics ( <<1). Third, we show the local stability of the equilibrium when the adaptive speed is very fast ( ). condition, We can obtain the Jacobian matrix (not shown) under the equilibrium (F1a) (F1b), (F1c). (F1d) For the special cases in which the trade-off functions are linear ( ) and the traits of the two species are the same ( condition, ), we obtain the local stability, (F2) where and. If, (F2) is always satisfied. This condition is likely to be satisfied if

12 . Next, we show the local stability criteria when the evolutionary dynamics is much slower than the population dynamics ( derivations in Appendix D. The local stability condition is <<1). The derivation is similar to the, (F3a), (F3b) where. Next, we study the conditions under which the above stability criteria are likely to be satisfied. When (A = 0), inequality (F3), is always satisfied if B 2 and C 2 < 0. When (A < 0), the criteria are likely to be satisfied if. When (A > 0), the criteria are likely to be satisfied if. In the special case in which the trade-off functions are linear (B 2, C 2 = 0), the criteria are simplified as follows,. (F4) When (A = 0), inequality (F4) is always satisfied. When (A < 0), (F4) is likely to be satisfied if. When (A > 0), (F4) is likely to be satisfied if. Finally we show the local stability of the equilibrium when the adaptive speed is very fast ( ). The derivation is similar to that in Appendix C. The conditions and can be rewritten as,, (F5a)

13 . (F5b) From (F5), we obtain the local stability condition,. (F6) Since at equilibrium, (F6) becomes. (F7) Using the equilibrium and, we obtain. (F8) This can be rewritten as ineq. (3) in the text.

14 Supplemental Figure Figure S1: The effect of the speed of adaptation on local stability when the trade-off functions are nonlinear (, < 0). We here adopted the functions r = r 0 (1 ) and g = g 0 (1 ). The parameter regions in which the equilibrium is unstable are shaded. (a) u * = v *. We assumed = 4. The small panel above panel a shows an enlarged portion of that panel. (b) u * v *. We assumed = 4 and = 2. The dotted lines separate locally stable and from unstable regions, with the region on the left side being unstable and that on the other side being stable. The ordered pairs show the values of ( ) for each boundary. The shaded regions in panels a and b are unstable regions when = 4 and = 4, = 2, respectively. Other parameter values are r 0 = 1, g 0 = 1, a 0 = 3, = 20, and d = 0.2.

15 Supplemental Figure Figure S2: Effect of the nonlinear functional form of the capture rate. (a) = 10; (b) = 5. Shaded regions in panels a and b are unstable when = 4. The other parameter values are the same as in Fig. S1a.

16 Supplemental Figure Figure S3: The effect of the speed of adaptation on local stability. We here adopted the functions r = r 0 (1 ) and g = g 0 (1 ). (a) u * = v *. We assumed = 1.5. (b) u * v *. We assumed = 1.7 and = 1.5 (K = 4). The dotted lines separate locally stable and from unstable regions, with the region on the inside being unstable and that on the outside being stable. The numbers in panel a show the values of K for each boundary. The shaded region in panel is unstable regions when K = 3. The ordered pairs in panel b show the values of ( ) for each boundary. Other parameter values are r 0 = 1, g 0 = 1, a 0 = 1, = 10, and d = 0.05.

17 Supplemental Figure Figure S4: Bifurcation diagrams of population (a, b) and trait dynamics (c, d) and nonequilibrium dynamics (e) in the two species in relation to the speed of adaptation. The points indicate the minimum and maximum values. We assumed K = 6. Other parameter values are same with those of Fig. S3. The initial values are (X, Y, u, v) = (1, 0.5, 0.1, 0.1).