Eco-Evolutionary Dynamics: Feedback & Oscillations

• Eco-evolutionary dynamics are defined as the reciprocal interactions between evolutionary traits and ecological processes, where strategic changes affect resource levels and environmental quality.
• Mathematical models using replicator equations and feedback-coupled bimatrix games reveal conditions for sustained oscillations in strategy abundance.
• These models inform practical interventions in microbial cooperation and social-ecological systems to mitigate overexploitation and stabilize population cycles.

Eco-evolutionary dynamics describe the coupled and reciprocal feedbacks between evolutionary change (strategic or trait dynamics) and ecological processes (resource renewal, population density, environmental state) on overlapping timescales. These dynamics are fundamental for systems in which strategy frequencies alter the ecological context (e.g., resource levels, environmental quality), which in turn modify the fitness or payoffs that determine evolutionary change. Typical applications include microbial cooperation, resource usage dilemmas, host-pathogen dynamics, and social-ecological systems subject to overexploitation or cyclical regime shifts.

1. Mathematical Formulation: Bimatrix Games with Feedback

The canonical eco-evolutionary model utilizes bimatrix games to encode interaction payoffs between two strategies (e.g., cooperation and defection). Each strategy pair $(i,j)$ receives a payoff defined by the entry $a_{ij}$ in a $2 \times 2$ matrix $A$. Critically, the payoff matrix itself is dynamically coupled to an environmental variable $s \in [0,1]$, representing the ecological state (such as resource abundance):

$A(s) = (1-s)A_0 + sA_1$

where $A_0, A_1$ are matrices corresponding to two distinct environmental regimes (e.g., depleted vs. abundant resource).

Population dynamics are governed by replicator equations with environmental feedback:

$\dot{x} = x(1-x)[f_C(x,s) - f_D(x,s)]$

$\dot{s} = \alpha x (1-s) - \beta (1-x) s$

where $x$ is the fraction of cooperators, $f_C/f_D$ are payoff-weighted average fitnesses, and $\alpha, \beta$ are forward/reverse ecological feedback rates.

2. Feedback Mechanisms and Environmental Dynamics

The eco-evolutionary model integrates environmental feedback as a dynamical variable modulated by population composition. Cooperators promote environmental improvement ($\alpha$) and defectors accelerate degradation ($\beta$), creating delayed negative feedback loops. Environmental change is typically implemented via a differential equation, but may also employ discrete switching laws:

$$s = \begin{cases} 0 & \text{if } x < x^* \ 1 & \text{if } x \ge x^* \end{cases}$$

for a threshold-based regime switch.

Instantaneous payoffs for each strategy become:

$$\begin{aligned} f_C &= a_{CC}(s)x + a_{CD}(s)(1-x) \ f_D &= a_{DC}(s)x + a_{DD}(s)(1-x) \end{aligned}$$

with

$a_{ij}(s) = (1-s)a_{ij} + s a'_{ij}$

.

3. Oscillatory Dynamics and Control via Feedback Switching

A principal result is that feedback-induced matrix switching enables persistent oscillations in strategy abundance (limit cycles or relaxation oscillations) even if neither environment supports stable coexistence. The mechanism involves cyclical dominance driven by delayed environmental restoration/degradation: defectors degrade the environment, which eventually penalizes them and favors cooperators; cooperators then recover the environment, restoring conditions that favor defectors, initiating new cycles.

Oscillation conditions require:

• Payoff matrices $A_0$, $A_1$ favor opposite strategies
• Sufficiently strong, timely environmental feedback
• Compatibility of ecological and evolutionary timescales

Formally, the coupled system is:

$$\begin{aligned} \dot{x} &= x(1-x)[f_C(x,s) - f_D(x,s)] \ \dot{s} &= \gamma (h(x) - s) \end{aligned}$$

where $h(x)$ specifies the environment’s "ideal" state given current strategy distribution and $\gamma$ is feedback speed.

4. Population State Control and Experimental Implications

Eco-evolutionary models with feedback-evolving games provide a foundation for population state control in microbial systems and social dilemmas. By externally adjusting environmental variables or feedback rates—effectively engineering the control law $g(x,s)$—experimenters can modulate the amplitude, frequency, or stability of population cycles. This enables:

• Accelerating or damping eco-evolutionary oscillations
• Delaying collapses in "tragedy of the commons" scenarios
• Designing interventions to stabilize cooperation or defection

These concepts are directly applicable to laboratory experiments with bacteria or yeast under resource limitation, and protocols may exploit designed environmental feedback for more robust population management.

5. Broader Applications in Social-Ecological Systems

The theoretical structure generalizes to domains where feedback between strategy and environment is mediated by shared resources (fisheries, forests, crowdsourcing platforms). By quantifying and controlling the feedback loops, social systems can design policies to deter overexploitation, promote sustainable use, or anticipate regime transitions. Mathematical analysis yields criteria for intervention efficacy and multi-stability, informing practical management.

6. Summary Table: Core Components and Conditions

Component	Mathematical Expression
Payoff Matrix	$A(s) = (1-s)A_0 + sA_1$
Replicator Equation	$\dot{x} = x(1-x)[f_C(x,s) - f_D(x,s)]$
Environmental Feedback	$\dot{s} = \alpha x(1-s) - \beta (1-x)s$
Switching Law	Discrete threshold: $s = 0$ if $x < x^*$, $s = 1$ if $x \ge x^*$
Oscillation Condition	Payoff matrices favor opposite strategies, strong feedback, compatible timescales

7. Relation to Previous Work and Context

This bimatrix feedback approach synthesizes recent results on feedback-evolving games (Shu et al., 2022), specifically building upon eco-evolutionary replicator models with environmental feedback (Gokhale et al., 2016, Wang et al., 2020), and generalized control laws for steering population dynamics (Wang et al., 2019). It extends the classic framework by allowing environmental structure to dynamically select among disjoint payoff regimes, offering a mechanism for sustained oscillations and more realistic eco-evolutionary trajectories observed in both experimental and natural populations.

• Weitz, J. S., et al., "An oscillating tragedy of the commons in replicator dynamics with game-environment feedback." Proc Natl Acad Sci USA, 113(47), E7518–E7525. (Montejano, 2016)
• Kaznatcheev, A., et al., "Evolving Interactions and Emergent Cycles in Rock-Paper-Scissors Games." PNAS, 114(13), E2572–E2578.

In summary, eco-evolutionary dynamics modeled by feedback-switching bimatrix games capture core features of real-world population cycles and dilemmas, providing rigorous criteria and analytical handles for predicting, controlling, and understanding coupled ecological-evolutionary processes.