The effect of reservoir-based chemical defense on predator-prey dynamics

Abstract

Numerous animal species use defense mechanisms such as chemical secretion to defend against attacks of predators. Although defense mechanisms have the potential to considerably change the dynamics and stability of a system, few theoretical studies exist. In this paper, we focus on predator-prey systems with reservoir-based chemical defense, which is also called “reducible defense” and is widespread among invertebrates. The predator has to attack often enough to disarm and consume prey, and prey can biosynthetically restore lost secretion. The model includes these features in the functional response, and in a separate equation for the stored amount of secretion. Additionally, our model takes into account that defense involves metabolic costs, reducing population growth of the prey. By performing computer simulations, we show that the defense mechanism causes the predator to take more time to consume prey. This time is increased more efficiently when the prey invests in a large reservoir rather than in fast restoration of secretion. We also investigate the stationary states resulting on longer time scales, finding that both predator and prey can become considerably more abundant due to the defense mechanism. However, investment into defenses pays off only when predator density is large enough and costs of defense are not too high.

Appendix A: Stochastic simulations

In order to test the quality of our expression for the fraction of defended prey D(S), we performed stochastic, individual-based simulations. The precise implementation and results are described in the following.

A.1 Methods

We consider a system with a single predator feeding on n prey individuals. Each prey individual has a secretion reservoir with a capacity K_S and a regeneration rate p_S. The regeneration of the amount of secretion S_i of the prey individual with index $i \in \{1,\dots ,n\}$ between attacks is given by the limited growth equation

$$\dot S_{i} = p_{S} \left( 1-\frac{S_{i}}{K_{S}} \right) . $$

If attacked, a prey individual defends itself against the attack by using an amount e_S of secretion out of the reservoir. In the case of a successful defense, the amount of secretion of the attacked individual is reduced by the transferring constant e_S. If there is not enough secretion in the reservoir (S_i < e_S), the attack succeeds. The predator performs attacks with a constant rate a. If an attack succeeds the predator has to wait for a handling time T_h before being able to perform its next attack. The attacked prey individual dies and is replaced by one with the mean amount of secretion of the prey population in order to keep the number of prey constant.

In order to perform the stochastic simulations, we use the Gillespie algorithm (Gillespie 1976). We initialize the system with filled reservoirs S_i(0) = K_S for $i \in \{1,\dots ,n\}$ at time t = 0. The Monte Carlo step is performed by determining the time Δt until the next attack using an exponential distribution with mean a^− 1 and by randomly assigning the target k of the attack. If the attack is successful, the handling time is added, Δt ↦ Δt + T_h, and the attacked individual dies and is replaced by a new prey individual (which obtains again the index k) with an amount of secretion corresponding to the population average

$$S_{k}(t + {\Delta} t) = \frac{1}{n}\sum\limits_{i = 1}^{n} S_{i}(t_{0} + {\Delta} t) . $$

If the attack is unsuccessful because the attacked prey individual can defend itself, its secretion is updated by

$$S_{k}(t + {\Delta} t) \mapsto S_{k}(t + {\Delta} t) - e_{S} . $$

Before the next Monte-Carlo step is performed, we update the amount of secretion for all individuals according to the limited growth equation,

$$S_{i}(t + {\Delta} t) = K_{S} - [K_{S} - S_{i}(t)]e^{-\frac{p_{S}}{K_{S}}{\Delta} t}, $$

for $i \in \{1,\dots ,n\}$. Then, we set the new time t ↦ t + Δt and perform the next Monte-Carlo step.

We run the simulation long enough and multiple times so that all possible reservoir filling levels occur often enough to obtain good statistics. We evaluate for each time step the proportion of defended prey D(S), which is the proportion of individuals with S_i(t) ≥ e_S, and the mean secretion S in the population. In order to get smooth curves, we divide the interval S ∈ [0, K_S] into discrete bins.

A.2 Results

Figure 5 shows the fraction of defended prey plotted against the mean amount of secretion of the prey population for different values of the transferring constant e_S. The results of the stochastic simulations are compared to the corresponding mean-field approximation defined in “Fraction of defended prey”. We used a population of 10 prey individuals and a fixed set a = 30, p_S = 0.02, K_S = 1, T_h = 0.6 for the remaining parameters, such that the ratios correspond to those in Table 1.

Fig. 5

The fraction of defended prey plotted against the mean amount of secretion for a population of 10 prey individuals. The parameters of the stochastic simulations are as follows: a = 30, p_S = 0.02, K_S = 1, T_h = 0.6. The parameter e_S is varied between the plots: ae_S = 0.05, be_S = 0.2, ce_S = 0.4, de_S = 0.6. In a, we used 100 bins in comparison to 50 bins in b, c, and d. In the plots linear interpolation is used in between the bins

In order to accommodate for the different number of time steps needed to reach equilibrium, we choose different numbers of runs for each set of parameters to achieve comparable statistics. We use in (a) 2000 runs with 10⁴ steps each, in (b) 1000 runs with 10⁴ steps each, in (c) 10⁴ runs with 1000 steps each, and in (d) 10⁵ runs with 100 steps each. In (a), we use double of the 10⁷ data points used in (b), (c), and (d) because we doubled the number of bins for increased precision in (a) in order to resolve the steeper slope.

The largest deviations between the stochastic results and the mean-field approximation can be found for relative small and large values of e_S, respectively (s. Fig. 5a, d with e_S = 0.05 and e_S = 0.6). Especially, for large values of e_S, a stepped growth of the fraction of defended prey with increasing amount of secretion S can be observed. For midrange values of e_S, the mean-field approximation fits rather well (s. Fig. 5c, d with e_S = 0.2 and e_S = 0.4).

In Fig. 6, we used the interpolated functions for the fraction of defended prey D(S) resulting from the stochastic simulations shown in Fig. 5 and compare the impact on the change over time of the consumption rate, the prey density, and the proportion of defended prey in dependence of the regeneration rate p_S, in analogy to the investigation in Fig. 1. For comparison, we added the results for the expression used in this paper (cp. Eq. 10) in the first row of Fig. 6a–c.

Fig. 6

The dynamics of prey consumption as a function of time and the regeneration rate p_S using the four versions to describe the fraction of defended prey D(S) shown in Fig. 5 (e_S = 0.05: (d-f); e_S = 0.2: (g-i); e_S = 0.4: (j-l); e_S = 0.6: (m-o)). For comparison, we added the results for the version used in the paper (a–c). The different color shades indicate prey consumption rate (a, d, g, j, m), prey density (b, e, h, k, n), and fraction of defended prey (c, f, i, l, o). The parameters that are not varied in a plot are set to the values given in Table 1

The exact overall form of the fraction of defended prey D(S) has just quantitative impact on the results. The less secretion is needed to defend a certain fraction of prey, the longer it takes to disarm (s. Fig. 6c, f, i, l, o) and to consume all prey (s. Fig. 6b, e, h, k, n). The time interval, the predators need to disarm prey to make it consumable (s. Fig. 6a, d, g, j, m) is higher the smaller the amount of secretion that suffices to arm all prey since the predators need to attack more often to make prey consumable.

In the case of a large transferring constant e_S, the stepped development of the fraction of defended prey D(S) has an impact on the development of the consumption rate with time; however, the general trends correspond to those observed with the mean-field function for the fraction of defended prey D(S) in “Effect of defenses on feeding rates”. Especially, the results for a transferring constant of e_S = 0.2, that we used in most of the paper, match those obtained with the mean-field approach pretty well (s. Fig. 6a–c, g–i).