Viewing communities as coupled oscillators: elementary forms from Lotka and Volterra to Kuramoto

Abstract

Ecosystems and their embedded ecological communities are almost always by definition collections of oscillating populations. This is apparent given the qualitative reality that oscillations emerge from consumer-resource interactions, which are the elementary building blocks for ecological communities. It is also likely always the case that oscillatory consumer-resource pairs will be connected to one another via trophic cross-feeding with shared resources or via competitive interactions among resources. Thus, one approach to understanding the dynamics of communities conceptualizes them as collections of oscillators coupled in various arrangements. Here we look to the pioneering work of Kuramoto on coupled oscillators and ask to what extent can his insights and approaches be translated to ecological systems. We explore the four ecologically significant coupling arrangements of the simple case of three oscillator systems with both the Kuramoto model and with the classical Lotka-Volterra equations. Our results show that the six-dimensional Lotka-Volterra systems behave strikingly similarly to that of the corresponding Kuramoto systems across all coupling combinations. This qualitative similarity in the results between these two approaches suggests that a vast literature on coupled oscillators may be relevant in furthering our understanding of ecosystem and community organization.

Appendix

$\frac{1}{2}\sum{\varvec{\Gamma}}=1$ for the Lotka-Volterra system of ordinary differential equations has C₁ with trophic-coupling on R₂ and R₃, C₂ with trophic-coupling on R₁, C₃ with trophic-coupling on R₁, and R₂ and R₃ resource coupled. Initial conditions for Fig. 2 are as follows: ${R}_{1}$ = 0.05, ${R}_{2}$ = 0.10, ${R}_{3}$ = 0.35, ${C}_{1}$ = 0.06, ${C}_{2}$ = 0.11, and ${C}_{3}$ = 0.36.

$$\dot{{R}_{1}}=b{R}_{1}\left(1-{R}_{1}\right)-\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\beta \frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-\beta \frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}$$

(4a)

$$\dot{{C}_{1}}=\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-m{C}_{1}+ \beta \frac{a{R}_{2}{C}_{1}}{1+ah{R}_{2}}+\beta \frac{a{R}_{3}{C}_{1}}{1+ah{R}_{3}}$$

(4b)

$$\dot{{R}_{2}}=b{R}_{2}\left(1-{R}_{2}-\alpha {R}_{3}\right)-\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-\beta \frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}$$

(4c)

$$\dot{{C}_{2}}=\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-m{C}_{2}+ \beta \frac{a{R}_{1}{C}_{2}}{1+ah{R}_{1}}$$

(4d)

$$\dot{{R}_{3}}=b{R}_{3}\left(1-{R}_{3}-\alpha {R}_{2}\right)-\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-\beta \frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}$$

(4e)

$$\dot{{C}_{3}}=\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-m{C}_{3}+ \beta \frac{a{R}_{1}{C}_{3}}{1+a{R}_{1}}$$

(4f)

for $\frac{1}{2}\sum\Gamma =-1$ for the Lotka-Volterra system of ordinary differential equations has C₁ with trophic-coupling on R₂; C₂ with trophic-coupling on R₁, R₂, and R₃ with resource-coupling; and R₁ and R₃ with resource-coupling. Initial conditions for Fig. 2 are as follows: ${R}_{1}$ = 0.60, ${R}_{2}$ = 0.30, ${R}_{3}$= 0.10, ${C}_{1}$ = 0.61, ${C}_{2}$ = 0.31, and ${C}_{3}$ = 0.11.

$$\dot{{R}_{1}}=b{R}_{1}\left(1-{R}_{1}-\alpha {R}_{3}\right)-\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\beta \frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}$$

(5a)

$$\dot{{C}_{1}}=\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-m{C}_{1}+ \beta \frac{a{R}_{2}{C}_{1}}{1+ah{R}_{2}}$$

(5b)

$$\dot{{R}_{2}}=b{R}_{2}\left(1-{R}_{2}-\alpha {R}_{3}\right)-\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-\beta \frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}$$

(5c)

$$\dot{{C}_{2}}=\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-m{C}_{2}+ \beta \frac{a{R}_{1}{C}_{2}}{1+ah{R}_{1}}$$

(5d)

$$\dot{{R}_{3}}=b{R}_{3}\left(1-{R}_{3}-\alpha {R}_{2}-\alpha {R}_{1}\right)-\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}$$

(5e)

$$\dot{{C}_{3}}=\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-m{C}_{3}$$

(5f)

for $\frac{1}{2}\sum{\varvec{\Gamma}}=-3$ has pairwise resource-coupling between all resources and no trophic-coupling. Initial conditions for Fig. 2 are as follows: ${R}_{1}$ = 0.50, ${R}_{2}$ = 0.55, ${R}_{3}$ = 0.45, ${C}_{1}$ = 0.51, ${C}_{2}$ = 0.56, and ${C}_{3}$ = 0.46.

$$\dot{{R}_{1}}=b{R}_{1}\left(1-{R}_{1}-{\alpha R}_{2} -{\alpha R}_{3}\right)-\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}$$

(6a)

$$\dot{{C}_{1}}=\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\mathrm{m}{C}_{1}$$

(6b)

$$\dot{{R}_{2}}=b{R}_{2}\left(1-{R}_{2}-{\alpha R}_{1} -{\alpha R}_{3}\right)-\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}$$

(6c)

$$\dot{{C}_{2}}=\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-\mathrm{m}{C}_{2}$$

(6d)

$$\dot{{R}_{3}}=b{R}_{3}\left(1-{R}_{3}-{\alpha R}_{1} -{\alpha R}_{2}\right)-\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}$$

(6e)

$$\dot{{C}_{3}}=\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-{\mathrm{m}C}_{3}$$

(6f)

for $\frac{1}{2}\sum\Gamma =3$ has pairwise trophic-coupling between all consumers and resources and no resource-coupling. Initial conditions for Fig. 2 are follows: ${R}_{1}$ = 0.10, ${R}_{2}$ = 0.20, ${R}_{3}$ = 0.45, ${C}_{1}$ = 0.11, ${C}_{2}$ = 0.21, and ${C}_{3}$= 0.46.

$$\dot{{R}_{1}}=b{R}_{1}\left(1-{R}_{1}\right)-\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\beta \frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}} -\beta \frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}$$

(7a)

$$\dot{{C}_{1}}=\frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-m{C}_{1}+ \beta \frac{a{R}_{2}{C}_{1}}{1+ah{R}_{2}} \, +\beta \frac{a{R}_{3}{C}_{1}}{1+ah{R}_{3}}$$

(7b)

$$\dot{{R}_{2}}=b{R}_{2}\left(1-{R}_{2}\right)-\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-\beta \frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\beta \frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}$$

(7c)

$$\dot{{C}_{2}}=\frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}-m{C}_{2}+ \beta \frac{a{R}_{1}{C}_{2}}{1+ah{R}_{1}} \, +\beta \frac{a{R}_{3}{C}_{2}}{1+ah{R}_{3}}$$

(7d)

$$\dot{{R}_{3}}=b{R}_{3}\left(1-{R}_{3}\right)-\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-\beta \frac{a{R}_{1}{C}_{1}}{1+ah{R}_{1}}-\beta \frac{a{R}_{2}{C}_{2}}{1+ah{R}_{2}}$$

(7e)

$$\dot{{C}_{3}}=\frac{a{R}_{3}{C}_{3}}{1+ah{R}_{3}}-m{C}_{3}+ \beta \frac{a{R}_{1}{C}_{3}}{1+a{R}_{1} } \, +\beta \frac{a{R}_{2}{C}_{3}}{1+ah{R}_{2}}$$

(7f)