Not attackable or not crackable—How pre‐ and post‐attack defenses with different competition costs affect prey coexistence and population dynamics

Abstract It is well‐known that prey species often face trade‐offs between defense against predation and competitiveness, enabling predator‐mediated coexistence. However, we lack an understanding of how the large variety of different defense traits with different competition costs affects coexistence and population dynamics. Our study focusses on two general defense mechanisms, that is, pre‐attack (e.g., camouflage) and post‐attack defenses (e.g., weaponry) that act at different phases of the predator—prey interaction. We consider a food web model with one predator, two prey types and one resource. One prey type is undefended, while the other one is pre‐ or post‐attack defended paying costs either by a higher half‐saturation constant for resource uptake or a lower maximum growth rate. We show that post‐attack defenses promote prey coexistence and stabilize the population dynamics more strongly than pre‐attack defenses by interfering with the predator's functional response: Because the predator spends time handling “noncrackable” prey, the undefended prey is indirectly facilitated. A high half‐saturation constant as defense costs promotes coexistence more and stabilizes the dynamics less than a low maximum growth rate. The former imposes high costs at low resource concentrations but allows for temporally high growth rates at predator‐induced resource peaks preventing the extinction of the defended prey. We evaluate the effects of the different defense mechanisms and costs on coexistence under different enrichment levels in order to vary the importance of bottom‐up and top‐down control of the prey community.


and invasion boundary
The calculation of coexistence equilibria, the linear stability analysis and the calculation of the invasion boundary are based on Jones and Ellner (2007). These calculations demand a rescaling of the model. At the end of the calculations, the traits are transferred back into their original units (prior to rescaling) to present the results in the main text.

Rescaling and reducing the dimensions of the predator-prey model
The analysed predator-prey model is given by In addition to the model presented in the main text (Eq. 2), we included also the digestion probability d i (see Appendix A). To allow for analytical calculations, we rescale the model with the following substitutions The rescaling ensures that all variables and parameters are dimensionless. The following equations represent the rescaled model The derivatives over time t ( dN dt , dA i dt , dB dt ) are transformed into derivatives over τ (Ṡ,ẋ i andẏ). The sum of the population densities ∑ = S + x 1 + x 2 + y is changing over time in dependence of the rate of change of its components∑ =Ṡ +ẋ 1 +ẋ 2 +ẏ (B4) Inserting the Eq. B3 into Eq. B4 results iṅ ∑ becomes zero when ∑ gets close to one implying that ∑ = 1 is an equilibrium (Jones and Ellner 2007). Thus, in the long-term, ∑ has a constant value of one meaning that the joint capacity of the populations S, x 1 , x 2 and y in the chemostat system is one. Considering long-term dynamics allows the following substitution for the resource density by inserting ∑ = 1 into ∑ = S + x 1 + x 2 + y and rearranging this equation Thus, the resource density can be represented by a function of the densities of the prey types and the predator (Eq. B6) allowing for a dimensional reduction of the model (Jones and Ellner 2007).
The rescaled, reduced model is represented by the following equationṡ One goal of this study is to find the conditions for coexistence of both prey types. Without a loss of generality, we explain our method based on the attack probability-maximum growth rate trade-off (p-β -TO) where the prey types do not differ in their consumption probability, digestion probability and their half-saturation constant, i.e., we assume that q 1 = q 2 = 1, d 1 = d 2 = 1 and k 1 = k 2 = k.
One condition for coexistence is the existence of an equilibrium (ẋ 1 ,ẋ 2 andẏ equal to zero) where all population densities are positive. This kind of equilibrium is called coexistence equilibrium (Jones and Ellner 2007). Equilibrium population densities are marked with a tilde. By insertinġ y = 0 into Eq. B7 the expression for the total attackable prey density at equilibrium ( is derived as The substitution S = 1 − x 1 − x 2 − y simplifies the further analysis (Eq. B6). The solution for the equilibrium predator density forẋ 1 = 0 is and forẋ 2 = 0 These solutions (Eq. B9 and B10) have to be equal. Equating both terms results in S corresponds to the equilibrium resource density. The total prey density in equilibrium X (= x 1 + x 2 ) is obtained by inserting S (Eq. B11) and y (Eq. B9 or B10) into X = 1 − S − y. As explained in Jones and Ellner (2007), the equilibrium prey population densities x 1 and x 2 can be represented as   x 1 Due to the coexistence condition stating that equilibrium population densities have to be positive ( x 1 and x 2 > 0) and remembering that p 1 < p 2 it follows from Equation B12 that p 1 X < Q < p 2 X (Jones and Ellner 2007). This term can be rearranged to The Inequation B13 defines the condition for the coexistence equilibrium where Q X can be considered as a density-weighted mean attack probability of the prey types. The attack probability of the defended prey is smaller and the attack probability of the undefended prey is larger than this density-weighed mean attack probability when there is a coexistence equilibrium.

Linear stability analysis
Another goal of this study is to identify the dynamics potentially occurring in this predator-prey system. A linear stability analysis enables to distinguish between locally stable equilibria (i.e. steady-state) and locally unstable equilibria (e.g. existence of stable limit cycles). Therefore, we analyse the local stability of the coexistence equilibrium. We refer here exemplary again to the p-β -TO and assume that d 1 = d 2 = 1 and k 1 = k 2 = k. The further steps are based on the study of Jones and Ellner (2007). The Jacobian matrix at the coexistence equilibrium J is generated by computing the necessary partial derivatives for the rescaled, reduced model (Eq. B7).
The roots of the characteristic polynomial of J are the eigenvalues λ of J. If all eigenvalues have negative real parts then the equilibrium is locally stable. The general equation of characteristic polynomials for a 3 × 3 dimensional matrix is The coefficients of the characteristic polynomial are The small letters in the definition of the coefficients represent the elements of the Jacobian matrix shown in Equation B14. The coefficient c 0 is equal to the negative determinant of J while c 2 equals the negative trace of J (Jones and Ellner 2007). The Routh-Hurwitz stability criterion (May 1974) is used for revealing whether the coexistence equilibrium is locally stable. According to this criterion, all eigenvalues (roots of the characteristic polynomial, Eq. B16) have negative real parts when the following conditions hold

Invasion boundary
Here we derive the invasion boundary of the defended prey A 1 invading a resident community of the undefended prey A 2 and the predator P. This demands no rescaling of the model. Hence, we refer to the original model (Eq. B1). We explain the calculation again based on the p-β -TO where q 1 = q 2 = 1, d 1 = d 2 = 1 and K 1 = K 2 = K. The invasion fitness of A 1 which is defined as the long-term mean per capita growth rate at very low densities (A 1 ≈ 0) is given by where long-term means are indicated by angle brackets. At the invasion boundary, the invasion fitness equals zero, i.e.
This can be rearranged to which yields the relationship between the maximum growth rate β 1 and the attack probability p 1 of the defended prey at its invasion boundary. The long-term means of the densities of the resident community N, A 2 and P (Eq. B21) have to be computed numerically over one cycle of the residents.