Mate Finding, Sexual Spore Production, and the Spread of Fungal Plant Parasites

Abstract

Sexual reproduction and dispersal are often coupled in organisms mixing sexual and asexual reproduction, such as fungi. The aim of this study is to evaluate the impact of mate limitation on the spreading speed of fungal plant parasites. Starting from a simple model with two coupled partial differential equations, we take advantage of the fact that we are interested in the dynamics over large spatial and temporal scales to reduce the model to a single equation. We obtain a simple expression for speed of spread, accounting for both sexual and asexual reproduction. Taking Black Sigatoka disease of banana plants as a case study, the model prediction is in close agreement with the actual spreading speed (100 km per year), whereas a similar model without mate limitation predicts a wave speed one order of magnitude greater. We discuss the implications of these results to control parasites in which sexual reproduction and dispersal are intrinsically coupled.

Appendix: Analogous Analysis Without Mate Limitation

In this section, sexual spores can be produced regardless of the presence of a mate, which amounts to removing the probability (i / 2) / n in Eq. (1). Without mate limitation, model (1) becomes

$$\begin{aligned} u_t= & {} \sigma i-\mu u+\kappa u_{xx}\,,\nonumber \\ i_t= & {} \left( q\mu u+p\alpha i\right) (n-i)/n\,. \end{aligned}$$

(15)

We rescale variables according to

$$\begin{aligned} t^*=\frac{t}{T}\,,\quad x^*=\frac{x}{L}\,,\quad i^*=\frac{i}{n}\,,\quad u^*=\frac{\mu }{\sigma n}\times u\,, \end{aligned}$$

and

$$\begin{aligned} a=\frac{1}{q\sigma T}\,,\quad \varepsilon =\frac{1}{\mu T}\,, \quad b=\frac{p\alpha }{q\sigma }\,,\quad d=\sqrt{\frac{\kappa }{\mu L^2}}\,. \end{aligned}$$

Dropping the asterisks for convenience, model (15) reads:

$$\begin{aligned} ai_t= & {} (u+ b i)(1-i)\,,\nonumber \\ \varepsilon u_t= & {} i - u+ d^2 u_{xx}\,. \end{aligned}$$

(16)

Applying the quasi-steady-state approximation to the second equation of (16) yields the following nonlocal integrodifferential equation for i:

$$\begin{aligned} ai_t=(1-i)\left( b i+\int _{-\infty }^{\infty } H(x-y)i(y,t)\,\mathrm {d}y\right) \,, \end{aligned}$$

(17)

where H is the Laplace kernel (6).

A Taylor expansion of i(y, t) at x in (17) yields

$$\begin{aligned} \int _{-\infty }^{\infty } H(x-y)i(y,t)\,\mathrm {d}y= & {} i(x,t)\int _{-\infty }^{\infty } H(x-y)\,\mathrm {d}y\nonumber \\&\quad +\, i_x(x,t)\int _{-\infty }^{\infty } (x-y)H(x-y)\,\mathrm {d}y\nonumber \\&\quad + \,\frac{i_{xx}(x,t)}{2}\int _{-\infty }^{\infty } (x-y)^2H(x-y)\,\mathrm {d}y\nonumber \\&\quad + \,\cdots \,. \end{aligned}$$

(18)

Using the moments of the Laplace distribution in (18), we get

$$\begin{aligned} ai_t = (1-i)(bi+i+d^2i_{xx}+d^4i_{xxxx}+\ldots )\,. \end{aligned}$$

(19)

A second-order Taylor expansion as d goes to zero in (19) yields

$$\begin{aligned} a i_t\approx (1-i)\left( b i+i+d^2i_{xx}\right) \,. \end{aligned}$$

(20)

Equation (20) can be expressed in a travelling wave form as

$$\begin{aligned} -\hat{c} I'=(1-I)\left( \left( b+1\right) I+I''\right) \,, \end{aligned}$$

(21)

where $\hat{c}=ac/d$, and the prime denotes differentiation w.r.t. $z=(x-ct)/d$. Proceeding as in Sect. 3.2, Eq. (21) can be expressed as a dynamical system:

$$\begin{aligned} \dot{I}= & {} J(1-I)\,,\nonumber \\ \dot{J}= & {} -{\hat{c}} J-(1-I)(b+1)I\,. \end{aligned}$$

(22)

Its equilibria are $(I,J)=(0,0)$ and (1, 0). Let G be the associated Jacobian matrix:

$$\begin{aligned} G=\left( \begin{array}{cc}-J&{}1-I\\ -(b+1)(1-2I)&{}-{\hat{c}}\end{array}\right) \,. \end{aligned}$$

Linearizing around (0, 0), we get

$$\begin{aligned} G_{(0,0)}=\left( \begin{array}{cc} 0&{}1\\ -(b+1)&{}-{\hat{c}}\end{array}\right) \,, \end{aligned}$$

whose eigenvalues are

$$\begin{aligned} \lambda _\pm =\frac{-{\hat{c}}\pm \sqrt{{\hat{c}}^2-4(b+1)}}{2}\,. \end{aligned}$$

So (0, 0) is a stable node if ${\hat{c}}>2\sqrt{b+1}$, and a stable spiral otherwise. From Murray (2002)’s Sect. 13.2 (Eq. 13.12 and Fig. 13.1), we conjecture that there is a trajectory from (1, 0) to (0, 0) lying entirely in the quadrant $I\ge 0$, $J\le 0$ with $0\le I\le 1$ for all wave speeds ${\hat{c}}\ge {\hat{c}}^\star$, with

$$\begin{aligned} {\hat{c}}^\star =2\sqrt{b+1}\,. \end{aligned}$$

We numerically checked that only ${\hat{c}}^\star$ is relevant, greater wave speeds being unstable. In terms of the original dimensional Eq. (15), the wave speed can be expressed as

$$\begin{aligned} c=q\sigma \sqrt{\frac{\kappa }{\mu }}2\sqrt{\frac{p\alpha }{q\sigma }+1}\,. \end{aligned}$$