The coupled dynamics of human socio-economic choice and lake water system: the interaction of two sources of nonlinearity

Abstract

We study a mathematical model for the coupled dynamics of human socio-economic choice and lake water system. In the model, many players choose one of the two options: a cooperative and costly option with low phosphorus discharge, and an economical option with high phosphorus discharge. The choice is affected by an economic cost, a social concern about water pollution, and a conformist tendency. The pollution level in the lake is determined by total phosphorus discharge by the players, the sedimentation and the outflow of phosphorus, and the recycling of phosphorus from the sediment. The model has two sources of nonlinearity: the cooperation level tends to be bistable due to conformist tendency of people (social hysteresis) and pollution level tends to be bistable because phosphorus recycling occurs faster in more eutrophic lakes (ecological hysteresis). The combination of these two sources may cause multiple stable equilibria or oscillations with a long periodicity. Small economic cost and strong social concern about pollution level can decrease the pollution level, but may not be very effective in enhancing the cooperation level. In contrast, strong conformist tendency produces a stable state with a high cooperation level and a low pollution level. We discuss implications of these results to the water quality management.

Appendix

Response to additional increase in recycling rate

The water pollution level can be lowered by a small difference of economic cost c or a large social concern κ, as well as by enhanced conformist tendency ξ. However, human activity may change parameter values in the dynamics of the pollution level in the lake. For example, water level control and bank protection works can modify the rate of limnological process, such as the recycling rate r. Then, we may ask which equilibrium is robust to additional nutrient load, such as a further enhancement of phosphorus recycling rate r.

Figure 8a and b illustrates that strong conformist tendency ξ makes the lake system robust to a further change of the recycling rate r. The isoclines for the pollution level are illustrated by broken line (r = 0.9) and bold broken line (r = 1), and the isoclines for the cooperation level are illustrated by solid line (ξ = 2) and bold solid line (ξ = 4). When r = 0.9 (I, broken line) and ξ = 2 (II, solid line), there is an unstable equilibrium A. When ξ increases to 4 (III, bold solid line), two stable equilibria (B, C) emerge. Fig. 8b shows the change of dynamics when ξ changes from 2 to 4 at time 1,000 (indicated by solid arrow), and r changes from 0.9 to 1 at time 2,000 (open allow). The cooperation level and the pollution level reach equilibrium C because the domain of attraction of equilibrium B is very small. The stable equilibrium C does not change when r changes to 1 (IV, bold broken line). The equilibrium of high cooperation level and low pollution level is robust to a further change of r (Fig. 8b).

Fig. 8

Response to additional increase in recycling r. a The isocline for the pollution level (broken line, bold broken line) and the isocline for the cooperation level (solid line, bold solid line). Isoclines with r = 0.9 (I, broken line), ξ = 2 (II, solid line), ξ = 4 (III, bold solid line), and r = 1 (IV, bold broken line). b The change of dynamics when ξ changes from 2 to 4 at time 1,000 (indicated by solid arrow), and r changes from 0.9 to 1 at time 2,000 (open arrow). c Isoclines with c = 8 (II, solid line), and c = 5.5 (III, bold solid line). The others are the same as in a. d The change of dynamics in the case of c. e Isoclines with c = 4 (III, bold solid line). The others are the same as in c. f The change of dynamics in the case of e. The standard set of parameters are: s = 0.01, γ = 2, c = 8, ξ = 2, κ = 1.6, α = 0.5, p _H = 0.12, p _L = 0.01, m = 1, and q = 2

In contrast, mildly reduced c cannot keep the pollution level low (Fig. 8c, d). When c is reduced from c = 8.0 (II) to c = 5.5 (III), cooperation level and pollution level reach stable equilibrium B of low pollution level and low cooperation level. However, increase of r (IV) arises stable equilibrium C of high pollution level. When r increases, the cooperation level and the pollution level jump from B to C (Fig. 8d). The equilibrium of low pollution level by mildly increased κ shows a similar response of the case of mildly reduced c.

A very large reduction of c or a very large enhance of κ can arise robust equilibrium B against the increase of r (Fig. 8e, f), but ξ seems to increase the cooperation level more than c and κ (Fig. 6). Hence, we consider that a greater conformist tendency ξ is more effective in restricting regime shift by additional recycling rate r than reduced c or enhanced κ.