Adaptive movement and food-chain dynamics: towards food-web theory without birth–death processes

Abstract

Population density can be affected by its prey [resource] and predator [consumer] abundances through two different mechanisms: the alternation of birth [or somatic growth] or death rate and inter-habitat movement. While the food-web theory has traditionally been built on the former mechanism, the latter mechanism has formed the basis of a successful theory explaining the spatial distribution of organisms in the context of behavioral and evolutionary ecology. Yet, few studies have compared these two mechanisms, leaving the question of how similar (or different) predictions derived from birth–death-based and movement-based food-web theories unanswered. Here, theoretical models of the tri-trophic (resource–consumer-top predator) food chain were used to compare food-web patterns arising from these two mechanisms. Specifically, we evaluated the response of the food-chain structure to inter-patch differences in productivity for movement-based models and birth–death-based models. Model analysis reveals that adaptive movements give rise to positively correlated responses of all trophic levels to increased productivity; however, this pattern was not observed in the corresponding birth–death-based model. The movement-based model predicts that the food chain response to productivity is determined by the sensitivity of animal movement to the environmental conditions. More specifically, increasing sensitivity of a consumer or top predator leads to smaller inter-patch variance of the resource or consumer density, while increasing inter-patch variance in the consumer or resource density. In conclusion, adaptive movement provides an alternative mechanism correlating the food-web structure to environmental conditions.

Appendix

I. Equilibrium state for the model with the birth–death process

• (a) Two-species model

The equilibrium is obtained by setting the right hand sides of Eq. 1a and b to zero and T to 0:

$$ r-\alpha {\displaystyle {R}^{\ast }}{\displaystyle {C}^{\ast }}=0 $$

(A1a)

$$ {e}_C\alpha {\displaystyle {R}^{\ast }}{\displaystyle {C}^{\ast }}-{d}_C{\displaystyle {C}^{\ast }}=0 $$

(A1b)

These equations are solved to give:

$$ R*=\frac{{\displaystyle {d}_C}}{{\displaystyle {e}_C}\alpha } $$

(A2a)

$$ {\displaystyle {C}^{\ast }}=\frac{r{\displaystyle {e}_C}}{{\displaystyle {d}_C}} $$

(A2b)

• (b) Three-species model

Similarly, the equilibrium for three-species is obtained by solving:

$$ r-\alpha {\displaystyle {R}^{\ast }}{\displaystyle {C}^{\ast }}=0 $$

(A3a)

$$ {e}_C\alpha {\displaystyle {R}^{\ast }}{\displaystyle {C}^{\ast }}-\beta {\displaystyle {C}^{\ast }}{\displaystyle {T}^{\ast }}-{d}_C{\displaystyle {C}^{\ast }}=0 $$

(A3b)

$$ {e}_T\beta {\displaystyle {C}^{\ast }}{\displaystyle {T}^{\ast }}-{d}_T{\displaystyle {T}^{\ast }}=0 $$

(A3c)

as:

$$ R*=\frac{r{\displaystyle {e}_T}\beta }{\alpha {\displaystyle {d}_T}} $$

(A4a)

$$ C*=\frac{d_T}{e_T\beta } $$

(A4b)

$$ T*=\frac{r{e}_C{e}_T}{{\displaystyle {d}_T}}-\frac{d_C}{\beta } $$

(A4c)

Note, this system is always locally stable. Under equilibrium conditions, we obtain the Jacobian matrix, as follows:

$$ J=\left(\begin{array}{ccc}\hfill -\frac{d_T\alpha }{e_T\beta}\hfill & \hfill -\frac{e_Tr\beta }{d_T}\hfill & \hfill 0\hfill \\ {}\hfill \frac{d_T{e}_C\alpha }{e_T\beta}\hfill & \hfill 0\hfill & \hfill -\frac{d_T}{e_T}\hfill \\ {}\hfill 0\hfill & \hfill \frac{e_T\left({e}_C{e}_Tr\beta -{d}_C{d}_T\right)}{d_T}\hfill & \hfill 0\hfill \end{array}\right) $$

(A5)

The characteristic equation, the solution of which is the eigenvalue, is given as:

$$ {\lambda}^3+{w}_1{\lambda}^2+{w}_2\lambda +{w}_3=0, $$

(A6)

where:

$$ {w}_1=\frac{d_T\alpha }{e_T\beta }, $$

(A7a)

$$ {w}_2={e}_Cr\left(\alpha +{e}_T\beta \right)-{d}_C{d}_T, $$

(A7b)

$$ {w}_3={e}_T\alpha \left(\frac{d_C{d}_T}{e_T\beta }-{e}_Tr\right). $$

(A7c)

The equilibrium point is locally stable if w ₁, w ₃ > 0 and w ₁ w ₂ > w ₃, according to the Routh-Hurwitz criteria. It is trivial that w ₁ > 0 because all parameter values are positive. Inequality w ₃ > 0 always holds, as long as the consumer has a positive equilibrium density. Hence, the stability condition reduces to w ₁ w ₂ > w ₃. We find the last condition reduces to rα > 0, which always holds.

II. Equilibrium state for the model with adaptive or non-adaptive movement

• (a) Two-species model with non-adaptive consumer

The equilibrium is obtained by solving:

$$ {r}_i-\alpha {\displaystyle {R}_i^{\ast }}{\displaystyle {C}_i^{\ast }}=0 $$

(A8a)

$$ 0.5{m}_C\left\{-{\displaystyle {C}_1^{\ast }}+{\displaystyle {C}_2^{\ast }}\right\}=0 $$

(A8b)

as:

$$ {\displaystyle {R}_i^{\ast }}=\frac{r_i}{\alpha {\displaystyle {C}_i^{\ast }}} $$

(A9a)

$$ {\displaystyle {C}_1^{\ast }}={\displaystyle {C}_2^{\ast }} $$

(A9b)

• (b) Two-species model with an adaptive consumer

The following equations should hold at equilibrium:

$$ {r}_i-\alpha {R}_i^{\ast }{C}_i^{\ast }=0 $$

(A10a)

$$ {m}_C\left\{-{f}_C{C}_1^{\ast }+\left(1-{f}_C\right){C}_2^{\ast}\right\}=0 $$

(A10b)

The two equations taken together give:

$$ \frac{{\displaystyle {C}_2^{\ast }}}{{\displaystyle {C}_1^{\ast }}}=\frac{1}{f_C}-1= \exp \left[{\theta}_C\left(\frac{r_2}{{\displaystyle {C}_2^{\ast }}}-\frac{r_1}{{\displaystyle {C}_1^{\ast }}}\right)\right], $$

(A11)

Note, when d(C ₁ + C ₂)/dt = 0, it holds that C ₁ + C ₂ = C _total, where C _total is constant. Thus, Eq. A8 becomes:

$$ \frac{C_{\mathrm{const}}}{{\displaystyle {C}_1^{\ast }}}-1= \exp \left[{\theta}_C\left(\frac{r_2}{C_{\mathrm{const}}-{\displaystyle {C}_1^{\ast }}}-\frac{r_1}{{\displaystyle {C}_1^{\ast }}}\right)\right]. $$

(A12)

The LHS of Eq. A12 is a monotonically decreasing function of C ₁ * with its limits:

$$ {\displaystyle \underset{{\displaystyle {C}_1^{\ast }}\to 0}{ \lim }}\left({C}_{\mathrm{const}}/{\displaystyle {C}_1^{\ast }}\right)-1=\infty $$

and

$$ {\displaystyle \underset{{\displaystyle {C}_1^{\ast }}\to {C}_{const}}{ \lim }}\left({C}_{const}/{\displaystyle {C}_1^{\ast }}\right)-1=0. $$

The RHS of Eq. A12 is a monotonically increasing function of C ₁ * with its limits:

$$ {\displaystyle \underset{{\displaystyle {C}_1^{*}}\to 0}{ \lim }} \exp \left[{\uptheta}_C\left\{{r}_2/\left({C}_{\mathrm{const}}-{\displaystyle {C}_1^{*}}\right)-{r}_1/{\displaystyle {C}_1^{*}}\right\}\right]=0 $$

and

$$ {\displaystyle \underset{{\displaystyle {C}_1^{*}}\to {C}_{\mathrm{const}}}{ \lim }} \exp \left[{\uptheta}_C\left\{{r}_2/\left({C}_{\mathrm{const}}-{\displaystyle {C}_1^{*}}\right)-{r}_1/{\displaystyle {C}_1^{*}}\right\}\right]=\infty . $$

Taken together, there should be unique ${\overline{C}}_1^{\ast }$, which holds Eq. A12.

$\overline{C_1^{\ast }}$ should also be larger than C _const/2 as follows. The LHS of Eq. A12 is 1 for C _i * = C _const/2, while the RHS is smaller than 1 for C ₁ * = C _const/2 as:

$$ {\theta}_C\left\{{r}_2/\left({C}_{\mathrm{const}}-{\displaystyle {C}_1^{\ast }}\right)-{r}_1/{\displaystyle {C}_1^{\ast }}\right\}<0 $$

suggesting that LHS should be larger than RHS for C ₁ * = C _const/2. Given that the LHS and LHS of Eq. A12 are decreasing and increasing functions of C ₁ *, respectively, the solution, $\overline{C_1^{\ast }}$, should be larger than C _const/2. Thus, it follows that C ₂ * > C ₁ * as:

$$ \overline{C_2^{\ast }}={C}_{\mathrm{const}}-\overline{C_1^{\ast }}<{C}_{\mathrm{const}}-{C}_{\mathrm{const}}/2={C}_{\mathrm{const}}/2<{C}_1^{\ast }. $$

Note, when $\left({C}_{\mathrm{const}}/\overline{C_1^{\ast }}\right)-1<1$, it should follow that:

$$ \exp \left[{\theta}_C\left\{{r}_2/\left({C}_{\mathrm{const}}-{\displaystyle {C}_1^{\ast }}\right)-{r}_1/{\displaystyle {C}_1^{\ast }}\right\}\right]<1 $$

suggesting that r ₂/C ^∗₂  − r ₁/C ^∗₁  < 0 and, thus, R ^∗₁  > R ^∗₂ .

• (c) Three-species model with a non-adaptive consumer and non-adaptive top predator

The following equations should hold at equilibrium:

$$ {r}_i-\alpha {R}_i^{\ast }{C}_i^{\ast }=0 $$

(A13a)

$$ 0.5{m}_C\left\{-{C}_1^{\ast }+{C}_2^{\ast}\right\}=0 $$

(A13b)

$$ 0.5{m}_T\left\{-{T}_1^{\ast }+{T}_2^{\ast}\right\}=0 $$

(A13c)

These three equations taken together give:

$$ {R}_i^{\ast }=\frac{r_i}{\alpha {\displaystyle {C}^{\ast }}} $$

(A14a)

$$ {\displaystyle {C}_1^{\ast }}={\displaystyle {C}_2^{\ast }}={\displaystyle {C}^{\ast }} $$

(A14b)

$$ {\displaystyle {T}_1^{*}}={\displaystyle {T}_2^{*}}={\displaystyle {T}^{*}} $$

(A14c)

It should hold that R ₁ * > R ₂ * as r ₁  > r ₂ .

• (d) Three-species model with a non-adaptive consumer and adaptive top predator

The following equations should hold at equilibrium:

$$ {r}_i-\alpha {R}_i^{\ast }{C}_i^{\ast }=0 $$

(A15a)

$$ 0.5{m}_C\left\{-{C}_1^{\ast }+{C}_2^{\ast}\right\}=0 $$

(A15b)

$$ {m}_T\left\{-{f}_T{T}_1^{\ast }+\left(1-{f}_T\right){T}_2^{\ast}\right\}=0 $$

(A15c)

Equations A15a and A15b give:

$$ {R}_i^{\ast }=\frac{r_i}{\alpha {\displaystyle {C}^{\ast }}} $$

(A16a)

$$ {\displaystyle {C}_1^{\ast }}={\displaystyle {C}_2^{\ast }}={\displaystyle {C}^{\ast }} $$

(A16b)

Further, note, when ${f}_T=\frac{1}{1+{e}^{\theta_T\left(\beta C-\beta C\right)}}=0.5$, it follows that:

$$ {\displaystyle {T}_1^{*}}={\displaystyle {T}_2^{*}}={\displaystyle {T}^{*}} $$

(A16c)

It should hold that R ₁ * > R ₂ * as r ₁  > r ₂ .

• (e) Three-species model with an adaptive consumer and non-adaptive top predator

The following equations should hold at equilibrium:

$$ {r}_i-\alpha {R}_i^{\ast }{C}_i^{\ast }=0 $$

(A17a)

$$ {m}_C\left\{-{f}_C{C}_1^{\ast }+\left(1-{f}_C\right){C}_2^{\ast}\right\}=0 $$

(A17b)

$$ 0.5{m}_C\left\{-{T}_1^{\ast }+{T}_2^{\ast}\right\}=0 $$

(A17c)

Equation A17c gives:

$$ {\displaystyle {T}_1^{*}}={\displaystyle {T}_2^{*}}={\displaystyle {T}^{*}} $$

(A18)

Using this equation with Eq. A17b, we get:

$$ {f}_C=\frac{1}{1+{e}^{\theta_C\left\{\left(\alpha {R}_i^{\ast }-\beta {\displaystyle {T}^{\ast }}\right)-\left(\alpha {R}_j^{\ast }-\beta {\displaystyle {T}^{\ast }}\right)\right\}}}=\frac{1}{1+{e}^{\theta_C\alpha \left({R}_i^{\ast }-{R}_j^{\ast}\right)}} $$

(A19)

Given this, Eqs. A17a and A17b are identical to A10a and A10b. Thus, R ^*₁  > R ^*₂ and C ^*₁  > C ^*₂ .

• (f) Three-species model with an adaptive consumer and adaptive top predator

At equilibrium, it should hold that:

$$ {r}_i-\alpha {R}_i^{\ast }{C}_i^{\ast }=0 $$

(A20a)

$$ {m}_C\left\{-{f}_C{C}_1^{\ast }+\left(1-{f}_C\right){C}_2^{\ast}\right\}=0 $$

(A20b)

$$ {m}_T\left\{-{f}_T{T}_1^{\ast }+\left(1-{f}_T\right){T}_2^{\ast}\right\}=0 $$

(A20c)

Using Eq. A13a, Eqs. A13b and A13c are transformed to:

$$ \frac{{\displaystyle {C}_2^{\ast }}}{{\displaystyle {C}_1^{\ast }}}=\frac{1}{f_C}-1= \exp \left[{\theta}_C\left\{\alpha \left(\frac{r_2}{{\displaystyle {C}_2^{\ast }}}-\frac{r_1}{{\displaystyle {C}_1^{\ast }}}\right)+\beta \left({\displaystyle {T}_1^{\ast }}-{\displaystyle {T}_2^{\ast }}\right)\right\}\right] $$

(A21a)

$$ \frac{{\displaystyle {T}_2^{\ast }}}{{\displaystyle {T}_1^{\ast }}}=\frac{1}{f_T}-1= \exp \left[{\theta}_T\left\{\alpha \left({\displaystyle {C}_2^{\ast }}-{\displaystyle {C}_1^{\ast }}\right)\right\}\right], $$

(A21b)

respectively. Note, when d(C ₁ + C ₂)/dt = 0 and d(T ₁ + T ₂)/dt = 0, it should hold that C ₁ + C ₂ = C _const and that T ₁ + T ₂ = T _const, where C _const and T _const are constants. Using these equations, Eqs. A13c and A14a taken together give:

$$ \frac{C_{\mathrm{const}}}{{\displaystyle {C}_1^{\ast }}}-1= \exp \left[{\theta}_C\left\{\alpha \left(\frac{r_2}{C_{\mathrm{const}}-{\displaystyle {C}_1^{\ast }}}-\frac{r_1}{{\displaystyle {C}_1^{\ast }}}\right)+\beta \left(\frac{2{T}_{\mathrm{const}}}{ \exp \left[{\theta}_T\alpha \left({C}_{\mathrm{const}}-2{\displaystyle {C}_1^{\ast }}\right)\right]+1}-{T}_{\mathrm{const}}\right)\right\}\right], $$

(A22)

LHS of Eq. A14b is a decreasing function of C ₁ * with its limits, ∞ (C ^∗₁  → 0) and 0(C ^∗₁  → C _const); RHS of Eq. A14b is an increasing function of C ₁ * with its limits, 0(C ^∗₁  → 0) and ∞ (C ^∗₁  → C _const). Therefore, there should exist a unique solution, C ^∗₁ , for Eq. A14b.

It should generally hold that T ^*₁  > T ^*₂ and, consequently, it should also hold that C ^*₁  > C ^*₂ and that R ^*₁  > R ^*₂ (r ₁/C ^*₁  > r ₂/C ^*₂ ). This is proved as follows. First, assume that T ^*₁  < T ^*₂ . Then, using Eq. A14b, it should follow that C ^*₁  < C ^*₂ . However, if both T ^*₁  < T ^*₂ and C ^*₁  < C ^*₂ hold, then Eq. A14a never holds, suggesting that it should not hold that T ^*₁  < T ^*₂ . Next, assume that T ^*₁  = T ^*₂ . Then, using Eq. A14b, it should follow that C ^*₁  = C ^*₂ . However, if both T ^*₁  = T ^*₂ and C ^*₁  = C ^*₂ hold, then Eq. A14a should never hold, suggesting that it should not hold that T ^*₁  = T ^*₂ . Consequently, it is proven that R ^*₁  > R ^*₂ , C ^*₁  > C ^*₂ and T ^*₁  > T ^*₂ always hold.