A general mathematical model for coevolutionary dynamics of mutualisms with partner discrimination

Abstract

Mutualisms (reciprocally beneficial heterospecific interactions) are thought as susceptible to exploitation by “cheaters” receiving benefits from partners without fair return. Theoretical studies suggest that partner discrimination is one of the key mechanisms to prevent cheaters from prevailing, but recently, a paradox is suggested that costly partner discrimination might collapse as the result of decreasing in quality variation among potential partners imposed by partner discrimination itself. Here, I develop a simple general mathematical model that consists of two host strains (discriminators/indiscriminators) and two symbiont strains (cooperators/non-cooperators) to establish a framework for the coevolutionary dynamics of mutualisms. First, I present a basic model, a positive equilibrium of which is neutrally stable. Secondly, I derive a formula to describe how the equilibrium shifts with a change in arbitrary parameters: I show a counter-intuitive result that the equilibrium frequency of discriminator hosts decreases as discrimination efficiency increases. Finally, I examine how the equilibrium and its stability changes by adding dependence of fitness of symbionts or hosts on their own frequencies: I find that negative or positive frequency dependence makes the equilibrium asymptotically stable or unstable, respectively. I also find that mutation and immigration of symbionts always make the equilibrium asymptotically stable, even if it does not increase low-quality symbionts.

Appendices

Appendix 1 Local stability of the equilibrium of the basic model

By the definition of equilibrium, φ = ψ = 0 at the equilibrium (H, S) = (H^*, S^*). In addition, I assume that the finesses of hosts and symbionts are independent from the frequencies of their own strains: ∂φ/∂H = ∂ψ/∂S = 0. From Eq. (4), I have

$$\frac{\partial V}{\partial H}=\frac{\psi }{H\left(1-H\right)}=0\;\mathrm{and}\;\frac{\partial V}{\partial S}=\frac{\varphi }{S\left(1-S\right)}=0$$

(22)

at the equilibrium (H^*, S^*). In addition,

$$\frac{{\partial }^{2}V}{\partial {H}^{2}}=\frac{{\psi }_{H}}{H\left(1-H\right)}-\frac{\left(1-2H\right)\psi }{{H}^{2}{\left(1-H\right)}^{2}}=\frac{{\psi }_{H}}{H\left(1-H\right)}>0,$$

$$\frac{{\partial }^{2}V}{\partial {S}^{2}}=-\frac{{\varphi }_{S}}{H\left(1-H\right)}+\frac{\left(1-2S\right)\varphi }{{S}^{2}{\left(1-S\right)}^{2}}=-\frac{{\varphi }_{S}}{H\left(1-H\right)}>0,$$

$$\frac{{\partial }^{2}V}{\partial H\partial S}=\frac{{\partial }^{2}V}{\partial S\partial H}=0,$$

because ψ_H = ∂ψ/∂H > 0 and φ_S = ∂φ/∂S < 0 at (H^*, S^*). Then

$$\frac{{\partial }^{2}V}{\partial {H}^{2}}\frac{{\partial }^{2}V}{\partial {S}^{2}}-{\left.\left(\frac{{\partial }^{2}V}{\partial H\partial S}\right.\right)}^{2}>0$$

(23)

Equations (22) and (23) indicate that V has a local minimum at (H^*, S^*). Moreover, from Eqs. (1), (2) and (4),

$$\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{\psi }{H\left(1-H\right)}\frac{\mathrm{d}H}{\mathrm{d}t}-\frac{\varphi }{S\left(1-S\right)}\frac{\mathrm{d}H}{\mathrm{d}t}$$

$$=\frac{\psi }{H\left(1-H\right)}H\left(1-H\right)\varphi -\frac{\varphi }{S\left(1-S\right)}S\left(1-S\right)\psi =0$$

(24)

at any points (H, S). Equation (24) indicates that the value of V is time invariant. Thus (H^*, S^*) is a neutrally stable equilibrium of Eqs. (1) and (2).

Appendix 2 Effects of frequency dependence on the position of the equilibrium

Let u be a continuous parameter for the degree of frequency dependence of symbiont strains; if u is positive (negative), the degree of the positive (negative) frequency dependence of symbiont strains monotonically increases with the magnitude of u. Functions ψ^FC and ψ^FN denote the frequency-dependent fitnesses of cooperator and non-cooperator symbionts, respectively. I assume that they are continuously partially differentiable with respect to u, and that ψ^FC = ψ^C and ψ^FN = ψ^N (and therefore ψ^F = ψ^C-ψ^N = ψ) when u = 0, where ψ^C and ψ^N are frequency-independent fitness functions of cooperator and non-cooperator symbionts, respectively.

The equilibrium of Eqs. (1) and (9) is denoted by (H^**, S^*). Note that H^** = H^* when u = 0. Applying the same procedure as Eqs. (5) and (6), I have

$$\frac{\mathrm{d}{H}^{**}}{\mathrm{d}u}=\frac{{\varphi }_{S}{\psi }_{u}^{F}-{\varphi }_{u}{\psi }_{s}^{F}}{{\varphi }_{H}{\psi }_{S}^{F}-{\varphi }_{S}{\psi }_{H}^{F}} \; \mathrm{and} \;\frac{\mathrm{d}{S}^{*}}{\mathrm{d}u}=\frac{{\varphi }_{u}{\psi }_{H}^{F}-{\varphi }_{H}{\psi }_{ u}^{F}}{{\varphi }_{H}{\psi }_{S}^{F}-{\varphi }_{S}{\psi }_{H}^{F}},$$

(25)

where ${\psi }_{H}^{F}=\partial {\psi }^{F}/\partial H$, ${\psi }_{S}^{F}=\partial {\psi }^{F}/\partial S$, ${\psi }_{u}^{F}=\partial {\psi }^{F}/\partial u$, and φ_u = ∂φ/∂u. As I have assumed φ_H = φ_u = 0, then at (H^**, S^*),

$$\frac{\mathrm{d}{H}^{**}}{\mathrm{d}u}=-\frac{{\psi }_{u}^{F}}{{\psi }_{H}^{F}}\mathrm{ and }\frac{\mathrm{d}{S}^{*}}{\mathrm{d}u}=0.$$

(26)

I have also assumed that ψ_H is positive at (H^*, S^*), the equilibrium of the frequency-independent system Eqs. (1) and (2). Therefore, if the magnitude of density dependence is sufficiently small, it should be ${\psi }_{H}^{F}>0$ at (H^**, S^*). Equation (26) indicates that the sign of $\mathrm{d}$ H^**/du is the opposite to the sign of ${\psi }_{u}^{F}$ at the equilibrium, which depends on the detail of the functions ψ^FC and ψ^FN.

To consider mutation of symbionts, I set k = μ_-/μ₊. Then Eq. (17) becomes

$${\psi }^{F}=\psi +{\mu }_{+}\left(\frac{1}{S}-\frac{k}{1-S}\right).$$

(27)

Equation (27) is continuously partially differentiable with respect to μ₊ and ψ^F = ψ when μ₊ = 0. Therefore, from Eq. (26),

$$\frac{\mathrm{d}{H}^{**}}{\mathrm{d}{\mu }_{+}}=-\frac{1}{{\psi }_{H}}\left(\frac{1}{{S}^{*}}-\frac{k}{1-{S}^{*}}\right).$$

(28)

Given ψ_H > 0 and ψ(H^*) = 0, Eq. (28) is followed by Eq. (16).

Similarly, I consider (sufficiently small) positive frequency dependence of host strains with replacing Eq. (2) with

$$\frac{\mathrm{d}H}{\mathrm{d}t}=H\left(1-H\right){\varphi }^{F},$$

(29)

where φ^F = φ^FD-φ^FI. In addition, I introduce a continuous parameter v and assume that φ^FD and φ^FI are continuously partially differentiable with respect to v, and that φ^FD = φ^D and φ^FI = φ^I when v = 0, where φ^D and φ^I are frequency-independent fitness functions of discriminator and indiscriminator hosts, respectively. A derivation similar to the above leads to

$$\frac{\mathrm{d}{H}^{*}}{\mathrm{d}v}=0\;\mathrm{and}\;\frac{\mathrm{d}{S}^{**}}{\mathrm{d}v}=-\frac{{\varphi }_{v}^{F}}{{\varphi }_{S}^{F}},$$

(30)

where (H^*, S^**) is the equilibrium of Eqs. (2) and (29). Given ${\varphi }_{S}^{F}<0$ at the equilibrium, Eq. (30) indicates that the sign of dS^**/dv is the same as the sign of ${\varphi }_{v}^{F}=\partial {\varphi }^{F}/\partial v$ at the equilibrium, which again depends on the detail of the functions φ^FD and φ^FI.

Appendix 3 Stability of the equilibrium and frequency dependence of host strains

Here I assume sufficiently small frequency dependence of host fitness on the frequency of its own strain in the population. The sign of the density dependence is the same as the sign of ∂φ^F/∂H. From Eqs. (2) and (29), I have the following components of the Jacobian matrix at the equilibrium (H^*, S^**):

$$\frac{\partial }{\partial H}\left(\frac{\mathrm{d}H}{\mathrm{d}t}\right)={H}^{*}\left(1-{H}^{*}\right)\frac{\partial {\varphi }^{F}}{\partial H},$$

(31)

$$\frac{\partial }{\partial S}\left(\frac{\mathrm{d}H}{\mathrm{d}t}\right)={H}^{*}\left(1-{H}^{*}\right)\frac{\partial {\varphi }^{F}}{\partial S}<0,$$

(32)

$$\frac{\partial }{\partial H}\left(\frac{\mathrm{d}S}{\mathrm{d}t}\right)={S}^{**}\left(1-{S}^{**}\right){\psi }_{H}>0,$$

(33)

$$\frac{\partial }{\partial H}\left(\frac{\mathrm{d}H}{\mathrm{d}t}\right)={H}^{*}\left(1-{H}^{*}\right)\frac{\partial {\varphi }^{F}}{\partial H},$$

(34)

$$\frac{\partial }{\partial S}\left(\frac{\mathrm{d}S}{\mathrm{d}t}\right)={S}^{**}\left(1-{S}^{**}\right)\frac{\partial \psi }{\partial S}=0.$$

(35)

If ∂φ^F/∂H > 0 (positive frequency dependence), it is found from Eqs. (31) and (34) that the trace of the Jacobian matrix.

$$\frac{\partial }{\partial H}\left(\frac{\mathrm{d}H}{\mathrm{d}t}\right)+\frac{\partial }{\partial S}\left(\frac{\mathrm{d}S}{\mathrm{d}t}\right)>0,$$

which indicates that the equilibrium is unstable. On the other hand, if ∂φ^F/∂H < 0 (negative frequency dependence), Eq. (14) is satisfied so that the equilibrium is asymptotically stable.