Invasion Dynamics in an Intraguild Predation System with Predator-Induced Defense

Abstract

Intraguild predation systems describe food webs in which an omnivorous predator competes with an intermediate prey for a basal resource. The classical intraguild predation system with Holling type II predation terms has the limitation of not being able to reproduce coexistence between predators in resource-rich environments despite its ubiquity in ecological systems. In this study, adaptive predator-induced and fitness-dependent defense of the intermediate predator is included into the model. In contrast to previous studies, this is done without an artificial bounding term. Numerical bifurcation software is used to show that adaptive defense mechanisms can significantly enhance parameter regimes leading to coexistence. Two different adaptation parameters are distinguished and linked to adaptations under different environmental conditions. The results indicate that the form of the reactivity–accuracy trade-off depends on the state of the environment. Finally, it is shown that an impact of adaptivity on dispersal abilities can considerably change shape and speed of invasion waves on a one-dimensional domain, which is important as those are the main measurable variables when examining data from biological invasions. The results indicate that a locally perfectly adaptive system can be globally (transient) maladaptive.

Appendices

Appendix A: Jacobian Matrix for the Model Without Adaptation

The Jacobian of the system without adaptation is given by

$$\begin{aligned} J = \begin{pmatrix} \alpha _{11} &{}\quad \alpha _{12} &{}\quad \alpha _{13}\\ \alpha _{21} &{}\quad \alpha _{22} &{}\quad \alpha _{23}\\ \alpha _{31} &{}\quad \alpha _{32} &{}\quad \alpha _{33} \end{pmatrix} \end{aligned}$$

(A.1)

with the entries

$$\begin{aligned} a_{11}&= r-\frac{2rR}{K}-\frac{\alpha _{ NR }N}{(1+\alpha _{ NR }H_{ NR }R)^2}-\frac{\alpha _{ PR }(P+\alpha _{ PN }H_{ PN }NP)}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2} \end{aligned}$$

(A.2a)

$$\begin{aligned} a_{12}&= R\left( \frac{\alpha _{ PN }\alpha _{ PR }H_{ PN }P}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2}-\frac{\alpha _{ NR }}{1+\alpha _{ NR }H_{ NR }R}\right) \end{aligned}$$

(A.2b)

$$\begin{aligned} a_{13}&= -\frac{\alpha _{ PR }R}{1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R} \end{aligned}$$

(A.2c)

$$\begin{aligned} a_{21}&= \frac{\alpha _{ NR }\epsilon _{ NR }N}{(1+\alpha _{ NR }H_{ NR }R)^2}+\frac{\alpha _{ PN }\alpha _{ PR }H_{ PR }NP}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2} \end{aligned}$$

(A.2d)

$$\begin{aligned} a_{22}&= \frac{\alpha _{ NR }\epsilon _{ NR }R}{1+\alpha _{ NR }H_{ NR }R}-m_N-\frac{P(\alpha _{ PN }+\alpha _{ PN }\alpha _{ PR }H_{ PR }R)}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2} \end{aligned}$$

(A.2e)

$$\begin{aligned} a_{23}&= -\frac{\alpha _{ PN }N}{1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R} \end{aligned}$$

(A.2f)

$$\begin{aligned} a_{31}&= \frac{\alpha _{ PR }(\epsilon _{ PR }+\alpha _{ PN }\epsilon _{ PR }H_{ PN }N-\alpha _{ PN }\epsilon _{ PN }H_{ PR }N)P}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2} \end{aligned}$$

(A.2g)

$$\begin{aligned} a_{32}&= \frac{\alpha _{ PN }P(\epsilon _{ PN }-\alpha _{ PR }\epsilon _{ PR }H_{ PN }R+\alpha _{ PR }\epsilon _{ PN }H_{ PR }R)}{(1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R)^2} \end{aligned}$$

(A.2h)

$$\begin{aligned} a_{33}&= \frac{\alpha _{ PN }\epsilon _{ PN }N+\alpha _{ PR }\epsilon _{ PR }R}{1+\alpha _{ PN }H_{ PN }N+\alpha _{ PR }H_{ PR }R}-m_P. \end{aligned}$$

(A.2i)

Appendix B: Nullclines of the Model with Adaptation

The nullcline for the model with adaptation can be written as

$$\begin{aligned} P&= \frac{n_b}{d_b} \end{aligned}$$

with

$$\begin{aligned} n_b =&2bm_N(\alpha +b+\alpha \alpha _{ PN }H_{ PN }N\nonumber \\&+\,\alpha _{ PN }H_{ PN }bN+\alpha _{ PN }H_{ PN }I_\mathrm{{max}}bN+\alpha _{ PR }(a+b)H_{ PR }R)^2 \end{aligned}$$

(B.1a)

and

$$\begin{aligned} d_b&= \alpha \alpha _{ PN }^2H_{ PN }I_\mathrm{{max}}N;\\ P&= \frac{n_R}{d_R} \end{aligned}$$

with

$$\begin{aligned} n_R =&-\,(\alpha +b+\alpha \alpha _{ PN }H_{ PN }N+\alpha _{ PN }H_{ PN }bN+\alpha _{ PN }H_{ PN }I_\mathrm{{max}}bN\nonumber \\&+\,\alpha _{ PR }H_{ PR }(\alpha +b)R)(\alpha _{ NR }KN-(K-R)(1+\alpha _{ NR }H_{ NR }R)r) \end{aligned}$$

(B.1b)

and

$$\begin{aligned} d_R&= \alpha _{ PR }K(\alpha +b)(1+\alpha _{ NR }H_{ NR }R);\\ N&= \frac{n_N}{d_N} \end{aligned}$$

with

$$\begin{aligned} n_N =&-\,(a+b)((1+b^2)m_N(1+\alpha _{ NR }H_{ NR }R)(1+\alpha _{ PR }H_{ PR }R)\\&+\,\alpha _{ PN }(P+\alpha _{ NR }H_{ NR }PR))\\&-\,\alpha _{ NR }\epsilon _{ NR }R(1+\alpha _{ PR }H_{ PR }R)) \end{aligned}$$

and

$$\begin{aligned} d_N&= \alpha _{ PN }H_{ PN }(\alpha +b+bI_\mathrm{{max}})(-\alpha _{ NR }\epsilon _{ NR }R+m_N(1+b^2)(1+\alpha _{ NR }H_{ NR }R)); \end{aligned}$$

(B.1c)

$$\begin{aligned} N&= \frac{n_P}{d_P} \end{aligned}$$

with

$$\begin{aligned} n_P&= (\alpha +b)(m_P-\alpha _{ PR }\epsilon _{ PR }R+\alpha _{ PR }H_{ PR }m_PR) \end{aligned}$$

(B.1d)

and

$$\begin{aligned} d_P&= \alpha _{ PN }\epsilon _{ PN }(\alpha +b)-\alpha _{ PN }H_{ PN }m_P(\alpha +b+I_\mathrm{{max}}b); \end{aligned}$$

for the differential equations modeling the dynamics of the adaptation state, the resource, the IGprey and the IGpredator, respectively.