Evolutionary games with environmental feedbacks

Abstract

Strategic interactions arise in all domains of life. This form of competition often plays out in dynamically changing environments. The strategies employed in a population may alter the state of the environment, which may in turn feedback to change the incentive structure of strategic interactions. Feedbacks between strategies and the environment are common in social-ecological systems, evolutionary-ecological systems, and even psychological-economic systems. Here we develop a framework of 'eco-evolutionary game theory' that enables the study of strategic and environmental dynamics with feedbacks. We consider environments governed either by intrinsic growth, decay, or tipping points. We show how the joint dynamics of strategies and the environment depend on the incentives for individuals to lead or follow behavioral changes, and on the relative speed of environmental versus strategic change. Our analysis unites dynamical phenomena that occur in settings as diverse as human decision-making, plant nutrient acquisition, and resource harvesting. We discuss implications in fields ranging from ecology to economics.

Fig. 1. A graphical illustration of how incentive parameters in eco-evolutionary games influence dynamics.

The horizontal axis of the state space corresponds to the frequency of individuals using the strategy with low impact on the environment, whereas the vertical axis indicates the quality of the environment, n, with the dashed line representing the attracting environmental nullcline. Each of the four incentive parameters (δ's and Δ's) control the direction and magnitude of strategy dynamics at a corner of the state space: strategy dynamics follow the red arrows when the corresponding δ or Δ is positive, and blue arrows when negative. When all are positive, meaning there are incentives to lead and to follow strategic changes, then some form of cyclical dynamics seem plausible. However, we show that all δ's and Δ's being positive is neither necessary nor sufficient for cyclic dynamics in eco-evolutionary games.

Fig. 2. Dynamical outcomes of linear eco-evolutionary games.

Each panel shows the dynamical outcomes for different regimes of incentives to lead strategic change, ${\mathrm{\Delta }}_{\mathrm{L}}^1$ and ${\delta }_{\mathrm{H}}^0$. Yellow regions denote parameter regimes that can produce limit cycles, provided ϵ < ϵ_crit, with level curves of ϵ_crit shown as black lines. Blue regions represent regions with a single dynamical outcome. Green regions represent bistability. a Outcomes when the incentives to lead strategic change are both negative. b Outcomes when incentives to lead strategic change are both positive. c, d Outcomes when incentives to lead strategic change are mixed. In these cases the yellow regions can exhibit bistability, dominance by one strategy, or cycles that occur in a bistable regime; the value of ϵ determines which of these outcomes occur. “Renewable resource system-level analysis” in Supplementary Note 1 provides analytical expressions for the boundaries between these dynamical regimes, in terms of the incentive parameters ${\mathrm{\Delta }}_{\mathrm{H}}^1,{\delta }_{\mathrm{L}}^0,{\mathrm{\Delta }}_{\mathrm{L}}^1,{\delta }_{\mathrm{H}}^0$.

Fig. 3. Phase planes and temporal dynamics in eco-evolutionary games with a decaying resource.

Parameters are chosen to fall in the yellow region of Fig. 2b. Only the speed of environmental dynamics relative to strategy dynamics, ϵ, varies between the panels. a Convergence to an interior equilibrium occurs for high ϵ. b Convergence to a limit cycle occurs for low ϵ. ($\alpha =1,{\mathrm{\Delta }}_{\mathrm{L}}^1=2,{\delta }_{\mathrm{L}}^0=3,{\mathrm{\Delta }}_{\mathrm{H}}^1=1,{\delta }_{\mathrm{H}}^0=1$).

Fig. 4. Simulations of eco-evolutionary game dynamics from the regime with possible cycles in Fig. 2c.

In each panel, dynamics proceed in a counterclockwise direction and the color of the solution curves illustrate the basins of attraction. Green curves approach the state dominated by the low-impact strategy, and blue curves represent regions that approach either the interior equilibrium or a stable limit cycle. a System dynamics under fast environmental feedbacks. Here, the interior equilibrium is stable and has a large basin of attraction. b, c Cases with environmental dynamics of intermediate speed exhibit bistability between the low-impact dominated state and limit cycles in the interior of phase space. d Dynamics under slow environmental feedbacks. When feedback speed falls below a critical threshold, limit cycles are no longer possible and the entire phase space approaches the low-impact equilibrium. ($\alpha =1,{\mathrm{\Delta }}_{\mathrm{L}}^1=-1∕8,{\delta }_{\mathrm{L}}^0=1,{\mathrm{\Delta }}_{\mathrm{H}}^1=4,{\delta }_{\mathrm{H}}^0=2$).