Human-environment feedback and the consistency of proenvironmental behavior

Abstract

Addressing global environmental crises such as anthropogenic climate change requires the consistent adoption of proenvironmental behavior by a large part of a population. Here, we develop a mathematical model of a simple behavior-environment feedback loop to ask how the individual assessment of the environmental state combines with social interactions to influence the consistent adoption of proenvironmental behavior, and how this feeds back to the perceived environmental state. In this stochastic individual-based model, individuals can switch between two behaviors, 'active' (or actively proenvironmental) and 'baseline', differing in their perceived cost (higher for the active behavior) and environmental impact (lower for the active behavior). We show that the deterministic dynamics and the stochastic fluctuations of the system can be approximated by ordinary differential equations and a Ornstein-Uhlenbeck type process. By definition, the proenvironmental behavior is adopted consistently when, at population stationary state, its frequency is high and random fluctuations in frequency are small. We find that the combination of social and environmental feedbacks can promote the spread of costly proenvironmental behavior when neither, operating in isolation, would. To be adopted consistently, strong social pressure for proenvironmental action is necessary but not sufficient-social interactions must occur on a faster timescale compared to individual assessment, and the difference in environmental impact must be small. This simple model suggests a scenario to achieve large reductions in environmental impact, which involves incrementally more active and potentially more costly behavior being consistently adopted under increasing social pressure for proenvironmentalism.

Fig 1. Frequency of active behavior at equilibrium in the absence of environmental feedback (Eq (5)), with respect to the payoff differential (β) and social norm threshold (ν=δBδA+δB).

(a) Bistability occurs in the black filled area (depending on the initial conditions, the equilibrium is either x* = 0 or x* = 1). (b) The upper equilibrium value (x* = 1) is plotted across the bistability area (reachable for initial frequency x₀ > ν). (c) The lower equilibrium value (x* = 0) is plotted across the bistability area (reachable for initial frequency x₀ < ν). Environmental sensitivity is ℓ = 0.1 and other parameters are set to their default values (Table 1).

Fig 2. Frequency of active behavior at equilibrium in the presence of environmental feedback (Eq (16)), with respect to the payoff differential (β) and social norm threshold (ν=δBδA+δB), for low to high individual sentivity to the environment (τ) (for (a), (d) and (g) ℓ = 0.1) and environmental reactivity (ℓ).

The place of each panel (a)-(i) gives the values taken for τ and ℓ = 0.25. For example, for (e), ℓ = 0.25 and τ = 1. Bistability occurs in the black filled areas. Stable limit cycles occur in the red filled areas. The environmental impact differential is fixed (l_B = 1, l_A = 0.7). Other parameters (κ, δ_B) are set to their default values (Table 1).

Fig 3. Total switching rate at stationary state.

The total switching rate is equalt to the variance of the asymptotic fluctuations around the equilibrium x* as given by Eq 18. (a) Variance for τ = 0.1. (b) Variance for τ = 1. (c) Variance for τ = 10. The environmental impact differential is fixed (l_B = 1, l_A = 0.7), environmental sensitivity is ℓ = 0.1 and other parameters are set to their default values (Table 1).

Fig 4. Influence of the environmental impact differential, l_B − l_A, on the frequency of active behavior (a, b, c), perceived environmental state (d, e, f), and total switching rate (g, h, i) at equilibrium.

For (a), (d) and (g), the parameters are l_B = 1 and l_A = 0.1. For (b), (e) and (h), l_B = 1 and l_A = 0.95. For (c), (f) and (i), l_B = 0.15 and l_A = 0.1. Individual sensitivity to the environment and environmental reactivity are set to τ = 1 and ℓ = 0.1. Other parameters are set to their default values (Table 1).

Fig 5. Effect of stochastic fluctuations on behavior-environment dynamics in the bistable case.

(a) Convergence of two trajectories issued from the same initial condition to alternate equilibria. (b) A single trajectory, with color coding for passing time, visits alternate equilibria, from the higher x* (blue tones) to the lower x* (green) to the higher x* (orange) to the lower x* (red). The stochastic simulation algorithm is described in the Methods. Environmental sensitivity ℓ = 0.1 (as in Fig 2A), payoff difference β = −0.25 and social norm threshold ν = 0.3. Other parameters are set to their default values (Table 1).