Coupled ecological and social dynamics in a forested landscape: the deviation of individual decisions from the social optimum

Abstract

We present a Markov chain model for land-use dynamics in a forested landscape. This model emphasizes the importance of coupling socioeconomic and ecological processes underlying landscape change. We assume that a forest is composed of many land parcels, each of which is in one of a finite list of land-use states. The land-use state of each land parcel changes stochastically. The transition probability is determined by two processes: the forest succession and the decision of landowners. The landowner tends to choose the land-use state which has a high expected discounted utility, i.e., the sum of the current and the future utilities of the land parcel. Landowners take the likelihood of future landscape changes into account when making decisions. We focus on a three-state model in which forested, agricultural, and abandoned states are considered. The land-use composition at equilibrium was analyzed and compared with the social optimum that maximizes the net benefit of all landowners in a society. We show that when landowners make a myopic choice focused on short-term benefits, their individual decisions tend to push the entire landscape toward an agricultural state even if the forested state represents the highest utility. This land-use composition at equilibrium is very different from the social optimum. A long-term management perspective and an enhanced rate of forest recovery can eliminate the discrepancy.

Appendix

The expected discounted utility V and the transition matrix P are dependent on each other (Fig. 1)—the transition rate r _ij is a function of V _i and V _j (Eq. 2 in the text); V _i and V _j in turn depend on P (Eq. 3 in the text); but elements of P includes r _ij . We explain a method of recursive calculation that is performed to cope with the interdependence between V and P. Let P[V] be the transition matrix given the expected discounted utility V. We started with a simple set of the expected discounted utility, such as V ⁽⁰⁾=u in which there is no contribution of future utility. We then calculated the transition matrix P ⁽⁰⁾=P[V ⁽⁰⁾ ]. Given P ⁽⁰⁾, we obtained a set of the expected discounted utility ${\mathbf{V}}^{{(1)}} = {\sum\nolimits_{n = 0}^\infty {\omega ^{n} ({\mathbf{P}}^{{(0)}} )^{n} {\mathbf{u}}} }$ (see Eq. 3 in the text). As a next step, using V ⁽¹⁾, we calculated P ⁽¹⁾=P[V ⁽¹⁾ ], and then obtained ${\mathbf{V}}^{{(2)}} = {\sum\nolimits_{n = 0}^\infty {\omega ^{n} ({\mathbf{P}}^{{(1)}} )^{n} } }{\mathbf{u}}.$ We repeated this procedure, and when a series of V ⁽¹⁾, V ⁽²⁾, ... , V ⁽ⁿ⁾ converges (i.e., V ⁽ⁿ⁾=V ^{(n - 1)}), V and P satisfy both Eqs. 2 and 3 in the text.