Mate Limitation in Fungal Plant Parasites Can Lead to Cyclic Epidemics in Perennial Host Populations

Abstract

Fungal plant parasites represent a growing concern for biodiversity and food security. Most ascomycete species are capable of producing different types of infectious spores both asexually and sexually. Yet the contributions of both types of spores to epidemiological dynamics have still to been fully researched. Here we studied the effect of mate limitation in parasites which perform both sexual and asexual reproduction in the same host. Since mate limitation implies positive density dependence at low population density, we modeled the dynamics of such species with both density-dependent (sexual) and density-independent (asexual) transmission rates. A first simple SIR model incorporating these two types of transmission from the infected compartment, suggested that combining sexual and asexual spore production can generate persistently cyclic epidemics in a significant part of the parameter space. It was then confirmed that cyclic persistence could occur in realistic situations by parameterizing a more detailed model fitting the biology of the Black Sigatoka disease of banana, for which literature data are available. We discuss the implications of these results for research on and management of Sigatoka diseases of banana.

Appendix: Pair Formation, Mating Functions, and Mate Limitation

In Sect. 2.3, we derived a bilinear (quadratic under even sex ratio) mating function from first principles focusing on plant pathogenic fungi. In this section, we discuss whether alternate mating functions could be considered to account for mate limitation in fungal plant parasites in general.

According to Hadeler (2012), a good mating function $\phi (x,y)$, where x and y are densities associated with both mating types, should satisfy the following conditions:

• 1. Preservation of positivity: $\phi (x, 0) = \phi (0, y) = 0$ for all $x, y\ge 0$,

• 2. Homogeneity: $\phi (kx, ky) = k\phi (x, y)$ for $k \ge 0$,

• 3. Monotonicity: $u \ge 0$, $v \ge 0$ implies $\phi (x + u, y + v) \ge \phi (x, y)$ .

Possible mating functions which satisfy these criteria include

• the geometric mean: $\phi (x,y)=(xy)^{1/2}$,

• the harmonic mean: $\phi (x,y)=2xy/(x+y)$,

• the minimum: $\phi (x,y)=\min (x,y)$.

However, quoting Caswell (2001), “each of these function has been considered, and rejected by human demographers for one reason or another (Mc Farland 1975), but the harmonic mean is regarded as the less flawed.”

The bilinear mating function $\phi (x,y)=xy$ does not respect homogeneity since $\phi (kx,ky)=k^2xy$. Although this condition is indeed required for pair-formation, this does not invalidate our model since fungi and many other species with two mating types may not be monogamous, so that the mating function need not be restricted to pair formation.

Moreover, the homogeneity condition makes the function $\phi$ necessarily such that $\phi (x, x) = c x$ for some coefficient c (Hadeler 2012). In other words, such mating functions, per se, cannot account for mate limitation (or positive density dependence at low density), since the per capita mating rate is a constant c under even sex ratio. By contrast, the bilinear mating function yields a per capita mating rate with increase in population density: $\phi (x,x)/x=cx$ (Dennis 1989).

In fact, we believe that the bilinear (or quadratic under even sex ratio) mating function used in this study is the most natural one to model mate limitation, since it is simple and it can be derived from first principles; according to Dennis (1989), it was introduced by Volterra (1938). Note that Dennis (1989)’s caution that the quadratic (mate-limited) growth rate should not exceed the linear (mate unlimited) growth rate is easily taken into account by normalizing population density with respect to its carrying capacity, as naturally done in this study (i.e., in his notations, $\lambda \alpha n^2\le \lambda n$ is ensured by taking $\alpha =1/\bar{n}$, with $n\le \bar{n}$). Also, the bilinear mating function is commonly used in the literature, e.g., (Veit and Lewis 1996; Lehtonen and Kokko 2011).

Actually, we simply used a cubic equation similar to the classical $\dot{n}=kn(n-a)(1-n)$, where a represents an Allee effect threshold; this model is classically used to model mate-finding Allee effects in population dynamics (Lewis and Kareiva 1993). Indeed, with $\alpha =0$, our strictly sexual model reads $\dot{I}=\sigma I^2(N-I)$ (since in this case $S=N-I$), which amounts to taking $a=0$ in the former model.

Extending our study to a negative exponential (Eq. 2) or to a rectangular hyperbola (Dennis 1989) would be interesting, but we do not think this would qualitatively change the results.