Autochthonous or allochthonous resources determine the characteristic population dynamics of ecosystem engineers and their impacts

Abstract

Ecosystem engineering, or the modification of physical environments by organisms, can influence trophic interactions and thus food web dynamics. Although existing theory exclusively considers engineers using autochthonous resources, many empirical studies show that they often depend on allochthonous resources. By developing a simple mathematical model involving an ecosystem engineer that modifies the physical environment through its activities, its resource, and physical environment modified by the engineer, we compare the effects of autochthonous and allochthonous resources on the dynamics and stability of community with ecosystem engineers. To represent a variety of real situations, we consider engineers that alter either resource productivity, engineer feeding rate on the resource, or engineer mortality, and incorporate time-lagged responses of the physical environment. Our model shows that the effects of ecosystem engineering on community dynamics depend greatly on resource types. When the engineer consumes autochthonous resources, the community can exhibit oscillatory dynamics if the engineered environment affects engineer’s feeding rate or mortality. These cyclic behaviors are, however, stabilized by a slowly responding physical environment. When allochthonous resources are supplied as donor-controlled, on the other hand, the engineer population is unlikely to oscillate but instead can undergo unbounded growth if the engineered environment affects resource productivity or engineer mortality. This finding suggests that ecosystem engineers utilizing allochthonous resources may be more likely to reach high abundance and cause strong impacts on ecosystems. Our results highlight that community-based, compounding effects of trophic and physical biotic interactions of ecosystem engineers depend crucially on whether the engineers utilize autochthonous or allochthonous resources.

Appendix 1: Derivation of the scaled model

By explicitly incorporating the effect of physical environment (E) on the parameters K, a, and m, Eq. (1-3 ) can be modified as follows:

$$ \frac{dE}{dt}=eN-cE, $$

(8)

$$ \frac{dR}{dt}=g(R)-{a}_0\left(1+{f}_{\mathrm{a}}E\right)NR, $$

(9)

$$ \frac{dN}{dt}=N\left[b{a}_0\left(1+{f}_{\mathrm{a}}E\right)R-\frac{m_0}{1+{f}_{\mathrm{m}}E}\right], $$

(10)

where

$$ g(R)=\left\{\begin{array}{c}\hfill rR\left[1-\frac{R}{K_0\left(1+{f}_{\mathrm{K}}E\right)}\right]\hfill \\ {}\hfill r\left[{K}_0\left(1+{f}_{\mathrm{K}}E\right)-R\right]\hfill \end{array}\right.\begin{array}{c}\hfill \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em \mathrm{autochthonous}\kern0.5em \mathrm{r}\mathrm{esource},\hfill \\ {}\hfill \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em \mathrm{allochthonous}\kern0.5em \mathrm{r}\mathrm{esource}.\hfill \end{array} $$

(11)

By substituting t = t ^*/m ₀, E = erE ^*/(ca ₀), R = m ₀ R ^*/(ba ₀), and N = rN ^*/a ₀ into Eq. (8), we can obtain the following equations:

$$ \frac{dE\ast }{dt\ast }=\frac{c}{m_0}\left(N\ast -E\ast \right), $$

(12)

$$ \frac{dR\ast }{dt\ast }=\frac{r}{m_0}\left[\gamma \left(R\ast \right)-\left(1+\frac{f_{\mathrm{a}}erE\ast }{c{a}_0}\right)N\ast R\ast \right], $$

(13)

$$ \frac{dN\ast }{dt\ast }=N\ast \left[\left(1+\frac{f_{\mathrm{a}}erE\ast }{c{a}_0}\right)R\ast -\frac{c{a}_0}{c{a}_0+{f}_{\mathrm{m}}erE\ast}\right], $$

(14)

where

$$ \gamma \left(R*\right)=\left\{\begin{array}{c}\hfill R*\left[1-\frac{m_0c{a}_0R*}{b{a}_0{K}_0\left(c{a}_0+{f}_{\mathrm{K}}erE*\right)}\right]\mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em \mathrm{autochthonous}\kern0.5em \mathrm{r}\mathrm{esource},\hfill \\ {}\hfill \frac{b{a}_0{K}_0\left(c{a}_0-{f}_{\mathrm{K}}erE*\right)}{m_0c{a}_0}-R*\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em \mathrm{allochthonous}\kern0.5em \mathrm{r}\mathrm{esource}.\hfill \end{array}\right. $$

(15)

If we define parameters as v _E = c/m ₀, v _R = r/m ₀, κ ₀ = ba ₀ K ₀/m ₀, κ= κ₀(1 + φ _K E ^*), and φ _X  = f _X er/(ca ₀) (X = K, a, m), we can derive the scaled model in the main text.

Appendix 2: Local stability analysis for the engineering pathway through resource productivity

Here, we show that the equilibrium where the engineer persists is locally stable when the engineered environment increases resource productivity. In this appendix, we remove the asterisks of state variables for descriptive convenience. Because E _Q = N _Q and R _Q = 1, where X _Q (X = E, R, N) are abundances at the engineer-persistent equilibrium, the Jacobian matrix is written as

$$ \boldsymbol{J}=\left({v}_R\begin{array}{c}\hfill -{v}_E\hfill \\ {}\hfill \left({\left.\frac{\partial \gamma }{\partial E}\right|}_{E={N}_{\mathrm{Q}},R=1}\right)\hfill \\ {}\hfill 0\hfill \end{array}{v}_{\mathrm{R}}\begin{array}{c}\hfill 0\hfill \\ {}\hfill \left({\left.\frac{\partial \gamma }{\partial R}\right|}_{E={N}_{\mathrm{Q}},R=1}-{\left.\gamma \right|}_{E={N}_{\mathrm{Q}},R=1}\right)\begin{array}{c}\hfill {v}_E\hfill \\ {}\hfill -{v}_R\hfill \\ {}\hfill 0\hfill \end{array}\hfill \\ {}\hfill {N}_{\mathrm{Q}}\hfill \end{array}\right). $$

(16)

The Routh-Hurwitz conditions for local stability are that a ₁ > 0, a ₃ > 0, and a ₁ a ₂ − a ₃ > 0, where a _i (i = 1, 2, 3) are the coefficients of characteristic polynomial λ ³ + a ₁ λ ² + a ₂ λ + a ₃ = 0 (λ is an eigenvalue) for the Jacobian.

In the model with an autochthonous resource, γ = R [1 − R/κ ₀(1 + φ _K E)], and therefore

$$ {a}_1={\nu}_{\mathrm{E}}+\frac{\nu_{\mathrm{R}}}{\kappa_0\left(1+{\varphi}_{\mathrm{K}}{N}_{\mathrm{Q}}\right)}, $$

(17)

$$ {a}_2={\nu}_{\mathrm{R}}\left[{N}_{\mathrm{Q}}+\frac{\nu_{\mathrm{E}}}{\kappa_0\left(1+{\varphi}_{\mathrm{K}}{N}_{\mathrm{Q}}\right)}\right], $$

(18)

$$ {a}_3={\nu}_{\mathrm{E}}{\nu}_{\mathrm{R}}{N}_{\mathrm{Q}}\left[1-\frac{\varphi_{\mathrm{K}}}{\kappa_0{\left(1+{\varphi}_{\mathrm{K}}{N}_{\mathrm{Q}}\right)}^2}\right]. $$

(19)

Because a ₁ > 0 and

$$ {a}_1{a}_2-{a}_3=\frac{v_{\mathrm{R}}\left[{v}_{\mathrm{E}}{v}_{\mathrm{R}}+{\kappa}_0\left\{{v}_{\mathrm{E}}{\varphi}_{\mathrm{K}}+\left(1+{\varphi}_{\mathrm{K}}{N}_{\mathrm{Q}}\right)\left({v}_{\mathrm{E}}^2+{v}_{\mathrm{R}}{N}_{\mathrm{Q}}\right)\right\}\right]}{\kappa_0^2{\left(1+{\varphi}_{\mathrm{K}}{N}_{\mathrm{Q}}\right)}^2}>0, $$

(20)

local stability is determined by the sign of a ₃, i.e., Θ = 1 − φ _K /[κ ₀(1 + φ _K N _Q)²], where

$$ {N}_{\mathrm{Q}}=\frac{\varphi_{\mathrm{K}}-1+\sqrt{{\left({\varphi}_{\mathrm{K}}-1\right)}^2+4{\varphi}_{\mathrm{K}}{\kappa}_0\left({\kappa}_0-1\right)}}{2{\varphi}_{\mathrm{K}}{\kappa}_0}. $$

(21)

Since N _Q is an increasing function of κ ₀, Θ is also an increasing function of κ ₀. Conditions for the occurrence of engineer-persistent equilibrium are κ ₀ > 1 if φ _K< 1 and κ ₀ > 4φ _K/(1 + φ _K)² if φ _K > 1. Therefore, Θ > 0 both if φ _K < 1 and if φ _K > 1, indicating that the equilibrium is locally stable as long as it exists.

In the model with an allochthonous resource, as γ = κ ₀(1 + φ _K E) − R, the coefficients of characteristic polynomial are that a ₁  = v _E  + v _R(1 + N _Q) > 0, a ₂  = v _R[v _E(1 + N _Q) + N _Q], and a ₃  = v _E v _R N _Q(1 − φ _K κ ₀). The condition for local stability a ₃  > 0 is satisfied because a condition for the occurrence of engineer-persistent equilibrium is κ ₀ < 1/φ _K. Moreover, a ₁ a ₂ − a ₃ = v _R[v _E(1 + N _Q){v _E  + v _R(1 + N _Q)} + v _R N _Q(1 + N _Q) + v _E φ _K κ ₀ N _Q] > 0. Thus, the engineering pathway through resource productivity makes the engineer-persistent equilibrium locally stable when the engineer uses an allochthonous resource as well as an autochthonous resource.