Appendix A: Stability analysis of a four-dimensional system
I studythe local stability of the system described by Eqs. (1) and (2) by linearizing the dynamics near the nontrivial equilibrium. I can judge the local stability by whether the characteristic equation of their Jacobian matrix satisfies the Routh-Hurwitz criteria.
I can obtain the Jacobian matrix (not shown) under the equilibrium condition,
(A1a)
(A1b)
(A1c)
(A1d)
where primes denote first derivatives. In the specific functionandand I find that
(A2)
at equilibrium. This result implies that at evolutionary equilibrium, the cost of increasing the value of the trait of one species must balance that of the other species. In other words, doubling the cost halves the population size at the equilibrium.
The characteristic equation for determining the eigenvalues is . The equilibrium point is locally stable if and , according to the Routh-Hurwitz criteria. Since the full stability condition is too complex to show, I show some limited cases. In addition, I assume r1= r2= r for the simplicity. When (= 0), the coefficients of the characteristic equation are
(A3a)
(A3b)
(A3c)
(A3d)
where ==andGT = +, andGP =. Hence, I have
(A4a)
(A4b)
(A4c)
The condition,, is not shown because of the complexity.
In thespecial case in which the trade-off functions are linear (= 0), the criteria are simplified as follows,
.(A5)
This corresponds to ineq. (3) in the text. Furthermore, when = = , I have
.(A6)
This means that a low interspecific competition (< 1) is necessary for stability. In addition, slow adaptation, low carrying capacity, and low cost of increasing trait are likely to stabilize the system.
Appendix B: Local stability analysis when coevolution is fast
I 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 (>1), both species should behave in a way that maximizes their fitness very quickly. Thus, I consider the local stability of equilibrium in aninterspecific competition system (1) for. The condition can be rewritten as,
(B1a)
(B1b)
From (B1), Iobtain
(B2)
The equilibrium of system (1) is locally stable if
(B3)
From (B2), the stability condition is
(B4)
Since < 0, Iobtain
.(B5)
This corresponds to ineq. (4) in the text.
Appendix C: 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:
(C1)
In this system, the population dynamics do not show cycles in the absence of evolutionary dynamics; thus, Ican use the equilibrium abundance instead of the mean abundance of one cycle.
Then (C1) can be rewritten as
(C2a)
(C2b)
By substituting the equilibrium values of the population dynamics, , while keeping trait values fixed, I obtain the evolutionary dynamics equations as follows,
(C3a)
.(C4b)
I study the local stability of this system. Under the equilibrium condition,
(C5a)
(C5b)
Ican evaluate the local stability by the signs of the trace and determinant of the Jacobian matrix (not shown). When (= 0), I obtain the stability condition:
(C6a)
(C6b)
where == and . I find that < 1 and 0 are likely to stabilize the system. These are correspond to ineq. (5) in the text.
In thespecial case in which the trade-off functions are linear (= 0), the criteria are simplified as follows,
(C7)
1