Mate search and mate-finding Allee effect: on modeling mating in sex-structured population models

Abstract

Many demographic and other factors are sex-specific. To assess their impacts on population dynamics, we need sex-structured models. Such models have been shown to produce results different from those predicted by asexual models, yet need to explicitly consider mating dynamics. Modeling mating is challenging and no generally accepted formulation exists. Mating is often impaired at low densities due to difficulties of individuals in locating mates, a phenomenon termed a mate-finding Allee effect. Widely applied models of this Allee effect assume either that only male density determines the rate at which females mate or that male and female densities are equal. Contrarily, when detailed models of mating dynamics are sometimes developed, the female mating rate is rarely reported, making quantification of the mate-finding Allee effect difficult. Here, we develop an individual-based model of mating dynamics that accounts for spatial search of one sex for another, and quantify the rate at which females mate, depending on male and female densities and under a number of reasonable mating scenarios. We find that this rate increases with male and female densities (hence observing a mate-finding Allee effect), in a decelerating or sigmoid way, that mating can be most efficient at either low or high female densities, and that the mate search rate may undergo density-dependent selection. We also show that mate search trajectories evolve to be as straight as possible when targets are sedentary, yet that when targets move the search can be less straight without seriously affecting the female mating rate. Some recommendations for modeling two-sex population dynamics are also provided.

Appendices

Appendix A: Continuous-time models of mating dynamics

Let each female encounter males at rate βm, for a positive constant β. This means that individual searchers scan a new area β each time unit. Moreover, the rate at which females mate equals βmf and thus follows the mass action law.

Scenario 1: Baseline

When males are not limited in their mating potential, the corresponding model of seasonal mating dynamics, assuming a large enough habitat area H², is

$$\begin{array}{@{}rcl@{}} \displaystyle{\frac{dm}{dt}} &=& 0, \\ \displaystyle{\frac{df}{dt}} &=& -\beta mf. \end{array} $$

(12)

Since male density m is constant, the density of unmated females declines exponentially as f(t) = f₀ exp(−βmt) for an initial female density f₀. Therefore, the proportion of females mated at the end of the mating season of length ϕ is

$$ P(\phi; m) = 1 - \exp(-\beta m \phi). $$

(13)

Scenario 2: Limited

Denoting by n _m the male mating potential (i.e., the maximum number of matings any male may have during the mating season) and by m _i , i = 1, …, n _m , the densities of males that have already mated i times, then males of class i go to class i + 1 at rate βm _i f, and become unavailable for mating after making n _m transitions (i.e., after mating n _m times). Therefore, under a limited male mating potential the baseline continuous-time model of mating dynamics (12) changes to

$$\begin{array}{@{}rcl@{}} && \displaystyle{\frac{dm}{dt}} = -\beta mf, \\ &&\displaystyle{\frac{df}{dt}} = -\beta mf - \beta m_{1} f - \beta m_{2} f - {\ldots} - \beta m_{n_{m}-1} f, \\ &&\displaystyle{\frac{dm_{1}}{dt}} = \beta mf - \beta m_{1} f, \\ &&\displaystyle{\frac{dm_{2}}{dt}} = \beta m_{1} f - \beta m_{2} f, \\ &&{\ldots} \\ &&\displaystyle{\frac{dm_{n_{m}-1}}{dt}} = \beta m_{n_{m}-2} f - \beta m_{n_{m}-1} f. \end{array} $$

(14)

We note that n _m = ∞ corresponds to unlimited polygyny (scenario 1). On the other hand, n _m = 1 corresponds to male monogamy and the model (14) then reduces to

$$\begin{array}{@{}rcl@{}} \displaystyle{\frac{dm}{dt}} &=& -\beta mf, \\ \displaystyle{\frac{df}{dt}} &=& -\beta mf. \end{array} $$

(15)

This model can be solved analytically. Indeed, Wells et al. (1990) showed that the female mating probability P depends on both male and female densities as

$$ P(\phi; m) = \left\{ \begin{array}{l} \displaystyle{\frac{m\exp[(m-f) \beta \phi]-m}{m\exp[(m-f) \beta \phi]-f} \text{\;\; if\;\;} m \neq f,} \\[2ex] \displaystyle{\frac{\beta m \phi}{1+\beta m \phi} \text{\; if\,\;} m = f.} \end{array} \right. $$

(16)

Scenario 3: Refractory

Denoting by m _s and m _r the densities of searching males and males currently in the refractory state, respectively, searching males enter the refractory class at the rate βm _s f and leave it at the rate m _r /T, where T is the (mean) length of refractory period. Therefore, modification of the baseline model (12) that accounts for the refractory period of males after each mating is

$$\begin{array}{@{}rcl@{}} && \displaystyle{\frac{dm_{s}}{dt}} = -\beta m_{s} f + \frac{1}{T}\,m_{r}, \\ &&\displaystyle{\frac{df}{dt}} = -\beta m_{s} f, \\ &&\displaystyle{\frac{dm_{r}}{dt}} = \beta m_{s} f - \frac{1}{T}\,m_{r}. \end{array} $$

(17)

Scenario 4: Learning

Denote by m _i and m _e the densities of inexperienced and experienced males, respectively. Both types of males search for females at rate β. Moreover, inexperienced and experienced males mate with probability p₀ and p₁ upon encounter, respectively. After n _t encounters with females, inexperienced males become experienced. With these assumptions, the baseline model (12) can be extended as

$$\begin{array}{@{}rcl@{}} &&\displaystyle{\frac{d{m_{i}^{1}}}{dt}} = -\beta {m_{i}^{1}} f, \\ &&\displaystyle{\frac{d{m_{i}^{k}}}{dt}} = \beta m_{i}^{k-1} f - \beta {m_{i}^{k}} f,\ k = 2,\ldots,n_{t} \\ &&\displaystyle{\frac{df}{dt}} = - p_{0} \beta \left( \sum\limits_{k = 1}^{n_{t}} {m_{i}^{k}} \right) f - p_{1} \beta m_{e} f, \\ &&\displaystyle{\frac{dm_{e}}{dt}} = \beta m_{i}^{n_{t}} f. \end{array} $$

(18)

By analogy with predator-prey theory and cooperative enzyme dynamics (see the main text), this learning scenario is expected to produce a sigmoid form of the female mating probability. While a sigmoid form is hardly seen to non-existent when only the male density m varies (Fig. 12a), it is clearly seen when both male and female densities change and are kept at a constant ratio (Fig. 12b).

Fig. 12

The mate-finding Allee effect for the learning scenario model (18) and for parameters β = 3, p₀ = 0, and p₁ = 1; the length of mating period ϕ = 4. Initial female density f = 0.08 in panel (a)

Scenario 5: Heterogeneity

The searchers may be heterogeneous in various ways. We return here to the baseline scenario and assume that males are not limited in their mating potential. Hence, each female’s mating probability at time t is P(t; m) = 1 − exp(−βmt). However, we let the parameter β differ among individual searchers, due to varying mate search rate. When females are the searching sex then the female mating probability becomes

$$ P_{F}(t; m) = {\int}_{0}^{\infty} (1 - \exp(-\beta m t)) f(\beta) d\beta, $$

(19)

where f(β) is a probability density function on β. On the other hand, when males are the searching sex then the female mating probability is

$$ P_{M}(t; m) = 1 - \exp\left( - m t {\int}_{0}^{\infty} \beta f(\beta) d\beta\right) = 1 - \exp(- \mu_{\beta} m t), $$

(20)

where μ _β is the mean value of β.

Assuming a Gamma-distributed search area β per unit time,

$$ f(\beta) = \frac{\theta^{\tau} \beta^{\tau-1}}{\Gamma(\tau)} \exp(-\theta \beta),\ \ 0<\beta<\infty, $$

(21)

and we get

$$ P_{M}(t; m) = 1 - \exp\left( - \frac{\tau}{\theta} m t\right) $$

(22)

and (Dennis 1989)

$$ P_{F}(t; m) = \frac{(\theta+mt)^{\tau}-\theta^{\tau}}{(\theta+mt)^{\tau}}. $$

(23)

Here, τ is a shape parameter and 𝜃 is a rate parameter of the Gamma distribution which are related to mean μ _β and variance var β as τ = (μ _β )²/var β and 𝜃 = μ _β /var β. For τ = 1, the Gamma distribution becomes an exponential distribution and P _F (t; m) = m/(𝜃 + m), one of the forms most commonly used to describe the female mating rate in population models accounting for a mate-finding Allee effect (Dennis 1989; Boukal and Berec 2002; Terry 2015). On the other hand, letting τ →∞ and 𝜃 →∞ such that τ/𝜃 → β the baseline model P(t; m) = 1 − exp(−βmt) is recovered (Dennis 1989).

Summary of Scenarios 1–5

Figure 13 summarizes how density of mated females increases in time or with male and female densities for the five examined mating scenarios. The monogamy scenario (i.e., the limited scenario with n _m = 1) is the worst scenario when the mating season is relatively long as the other scenarios allow males to mate multiply and hence increase the chance of females to meet at least one male. On the other hand, the learning scenario is the worst scenario if the mating season is relatively short as it catches up with male polygyny only when a majority of males are experienced. As we know from above, small refractory period and large male mating potential behave similarly to the baseline scenario with unlimited male polygyny. We also know from our individual-based simulations that heterogeneity in the mate search rate when females are the searching sex produces consistently lower female mating probabilities relative to the baseline scenario, while the results under heterogeneity in the mate search rate when males are the searching sex coincide with those for the baseline scenario.

Fig. 13

Dynamics of analytical models of mating dynamics corresponding to Scenarios 1–5. The heterogeneity scenario here considers mate search rate heterogeneity in females; mate search rate heterogeneity in males would give results identical to the baseline scenario. The female mating probability is plotted as a function of a time, with m₀ = f₀ = 0.2, and b male and female density, with ϕ = 4, for parameters β = 3, n _m = 2, T = 2, p₀ = 0, p₁ = 1, n _t = 2, and var β = 5

Appendix B: Derivation of a simple continuous-time two-sex model

Here, we exemplify a sex-structured population model in which the processes of mating, reproduction, and mortality occur simultaneously. We assume that when a male and a female encounter one another they mate. Upon mating, the female enters a gestation period of length t _F after which it gives birth to b offspring following a 1:1 sex ratio and then resumes mate search. Mated males are assumed to enter a refractory period of length T. Moreover, males and females die at rates m _M and m _F , respectively. We distinguish four state variables: searching males M _s and females F _s , and resting males M _r and females F _r . A continuous-time two-sex model may then be as follows:

$$\begin{array}{@{}rcl@{}} \displaystyle{\frac{dM_s}{dt}} &=& -\beta M_s F_s - m_M M_s + \frac{1}{T}M_r + \frac{b}{2} \frac{1}{t_F}F_r, \\ \displaystyle{\frac{dF_s}{dt}} &=& -\beta M_s F_s - m_F F_s + \frac{1}{t_F}F_r + \frac{b}{2} \frac{1}{t_F}F_r, \\ \displaystyle{\frac{dM_r}{dt}} &=& \beta M_s F_s - \frac{1}{T}M_r - m_M M_r, \\ \displaystyle{\frac{dF_r}{dt}} &=& \beta M_s F_s - \frac{1}{t_F}F_r - m_F F_r. \end{array} $$

(24)