Skiba points and heteroclinic bifurcations, with applications to the shallow lake system

Abstract

Techniques from dynamical systems, specifically from bifurcation theory, are used to investigate the occurrence of Skiba points in one-state, one-co-state control systems, for which the effect of the control has a definite direction. A Skiba point is an initial state for which two different optimal solutions of the control problem exist. It is found that the parameter region for which Skiba points occur is bounded by heteroclinic bifurcation manifolds. A local criterion is given that ensures the existence of Skiba points in systems with small discount rates. The analysis is applied to the shallow lake system investigated by Mäler et al. (in: K.-G. Mäler, C. Perrings (Eds.), The Economics of Non-Linear Dynamic Systems, Resilience Network, 2000, in preparation). For this system, it is shown that for any given parameter value, there is at most one Skiba point.

Introduction

This paper investigates the connection between heteroclinic bifurcations and the occurrence of Skiba points in one-dimensional optimal control problems. That is, optimal controls u(t) are sought, minimising a cost functional

$$\text{C}\text{=}\text{∫}\text{0}\text{∞}\text{g(x,u,λ)}\text{e}{}^{\mathrm{-\rho t}}\text{d}\text{t}$$

under the condition that x and u satisfy for all t the state equation

$$\text{x}\text{̇}\text{=f(x,u,λ).}$$

Here $\text{x∈X⊂}\text{R}$ denotes the state variable, $\text{u∈U⊂}\text{R}$ the control, and $\text{λ∈Λ⊂}\text{R}{}^{\mathrm{q}}$ a q-dimensional parameter. The state space X, the control space U, and the parameter space Λ are all assumed to be open sets. In this article, only the following class of state equations is investigated: it is assumed that the function f satisfies

$$\text{∂f}\text{∂u}\text{(x,u,λ)≠0}$$

for all (x,u,λ)∈X×U×Λ. Heuristically, this means that the control has an uni-directional effect.

It is well known that in this kind of system, there might exist so-called indifference or Skiba states, for which there are two distinct optimal controls with equal total cost. The central result of this article is that for this kind of system (modulo technical conditions outlined below), the set of parameters for which the control problem has Skiba points is bounded by the so-called heteroclinic bifurcation curves of the associated state–co-state system.

Skiba points occur naturally in control problems with several equilibria. The concept was introduced by Skiba (1978) for optimal grow paths of economies with a convex–concave production technology; subsequently indifference points have been found by Dechert and Nishimura (1983) and in many other places, see Deissenberg et al. (2001) and references therein.

Relation to bifurcation theory. Determining the existence of a point for which there are two equally costly solutions is typically a global problem: the cost functionals of two trajectories have to be computed. Now, typically a system depends on several parameters. Imagine the case that there is a set of parameters for which Skiba points exist, and another for which there are no Skiba points. There is a qualitative difference between these type of systems. Going from one set to the other hence entails a qualitative change, or, in technical terms, a bifurcation. It will be argued below that the simplest bifurcations that can occur for the one-state, one-control systems studied in this article are saddle-node and heteroclinic bifurcations. The former bifurcation is local, the latter global. Hence, it may be expected that heteroclinic bifurcations are in some way connected to the occurrence of Skiba points. As announced, the main result of this article is that this is indeed the case.

The significance of this result is that bifurcation theory can be used to determine the regions in parameter space for which Skiba points do or do not exist. For the shallow lake family of systems, a bifurcation diagram is given in Fig. 1.

Main application. The shallow lake family (see Brock and Starrett, 1999; Dechert and Brock, 2000; Mäler et al., 2000) shall serve as the main application of the general theory throughout the present article, since it has been the initial motivation for this study. Indeed, the first half of this article gives an analysis of the shallow lake system in terms of the general results obtained in the second half.

Biological model. The dynamics of pollution or eutrophication of shallow lakes gives rise to a simple optimal control problem which nevertheless has quite interesting features. For the general economic and ecological background of the model studied here, the reader is referred to Mäler et al. (2000), where the following model equation is introduced:

$$\text{x}\text{̇}\text{=f(x,u)=u−bx+}\text{x}{}^2\text{x}{}^2\text{+1}\text{,}\text{x(0)=x}{}_0\text{.}$$

Here x(t)⩾0 is proportional to the amount of phosphorus in a shallow lake, u(t)⩾0 to the input of more phosphorus (due to farming); b⩾0 is proportional to the rate of loss of phosphorus due to sedimentation, outflow and sequestration in other biomass. The term x²/(x²+1) models the biological production of phosphorus in the lake. These remarks are only intended to give an indication of the meaning of the variables; for more detailed information and references, see Mäler et al. (2000). Of course, there is much more to shallow lakes than the extremely simple model (4): see for instance Scheffer (1998).

In case there is more than one agent releasing phosphorus into the lake, the load u(t) will be the total sum of the individual loads u_i(t)

$$\text{u(t)=}\text{∑}\text{i=1}\text{n}\text{u}{}_{\mathrm{i}}\text{(t),}$$

where n denotes the number of agents.

Welfare. The welfare function, that is, the benefits to be reaped from the use of the lake, is modelled as

$$\text{log}\text{u−cx}{}^2\text{.}$$

Farmers use artificial fertilisers to grow crops; these fertilisers contain phosphorus, which in the end is washed into the lake. The term $\text{log}\text{u}$ models the benefits to farmers arising from using u units of phosphorus. The lake is also used by fishers and tourists, who are interested in a clean lake. The term −cx² represents the cost of pollution: here c⩾0 is an economic parameter. Future benefits are discounted by a factor e^−ρt, with ρ⩾0 the discount factor.

If u(t) is given and assumed to be continuous, except maybe for a set of isolated jump points, the solution $\text{x}\text{:}\text{R}{}_{\mathrm{+}}\text{→}\text{R}{}_{\mathrm{+}}$ is determined, and it will be continuous. The total benefits of the lake are then given by the welfare or benefit functional $\text{B}\text{[x,u]}$. It takes the functions x and u as its arguments, and it is given by

$$\text{B}\text{[x,u]=}\text{∫}\text{0}\text{∞}\text{(}\text{log}\text{u−cx}{}^2\text{)}\text{e}{}^{\mathrm{-\rho t}}\text{d}\text{t.}$$

The optimal control problem. Note that the dynamics of x are determined by Eq. (4), once the function u(t) has been chosen. The optimal control problem of a social planner is to find the optimal dumping level u such that $\text{B}$ is maximised. Equivalently, one can determine u to minimise the cost functional

$$\text{C}\text{[x,u]=}\text{∫}\text{0}\text{∞}\text{g(x,u)}\text{e}{}^{\mathrm{-\rho t}}\text{d}\text{t=}\text{∫}\text{0}\text{∞}\text{(−}\text{log}\text{u+cx}{}^2\text{)}\text{e}{}^{\mathrm{-\rho t}}\text{d}\text{t.}$$

Dynamic game. If several agricultural agents are using the lake—say, several countries border on the lake—every one of them has a separate cost functional $\text{C}{}_{\mathrm{i}}\text{[x,u]}$ to maximise. However, since the state of the lake is influenced by the actions of the other agents, these have to be taken into account as well. This defines a dynamic game. Now every agent has to choose a strategy: the game is said to be in Nash equilibrium if every agent's strategy, given the opponents’, is optimal. Because of the symmetry of the context, and because asymmetric equilibria are much harder to find, attention is restricted to symmetric equilibria, where every agent chooses the same strategy. In Mäler et al. (2000) it is shown that in the context of an n-agent game, an optimal strategy u_i(t) is equal to an optimal control of the one-agent problem with the parameter c replaced by c/n. Hence it suffices to restrict attention to the one-agent case.

Optimal solutions. The Pontryagin maximum principle yields a system of two differential equations for the state–co-state equations, whose phase curves correspond to potential optimal trajectories of the system. The actual optimal solution is selected from the continuum of solutions (x(t),u(t)) that satisfy x(0)=x₀, that are defined for all t∈[0,∞), and satisfy a transversality condition of the form

$$\text{lim}\text{t→∞}\text{u(t)}\text{e}{}^{\mathrm{\rho t}}\text{=∞.}$$

Hence, the control u(t) should remain bounded away from 0 as t→∞, or it should approach 0 not too quickly.

For positive ρ, the state–co-state system has positive divergence, which excludes the possibility of phase curves that form closed loops. By the Poincaré–Bendixon theorem (see Anosov et al., 1988), the phase portrait then consists only of equilibria and non-intersecting curves. Phase curves that are bounded as t→∞ are necessarily stable manifolds of hyperbolic saddle equilibria, or centre manifolds of non-hyperbolic (bifurcating) equilibria.

Skiba points. It turns out that for certain sets of parameters there are initial states $\text{x=x}{}_{\mathrm{\ast }}$, such that there are optimal controls u₁(t) and u₂(t), and corresponding x₁(t), x₂(t), ($\text{x}{}_1\text{(0)=x}{}_2\text{(0)=x}{}_{\mathrm{\ast }}$), such that

$$\text{C}\text{[x}{}_1\text{,u}{}_1\text{]=}\text{C}\text{[x}{}_2\text{,u}{}_2\text{].}$$

The initial state $\text{x}{}_{\mathrm{\ast }}$ is then called an indifference or Skiba point.

The first part of the article sketches the main ideas in the case of the shallow lake system. Specifically, for parameter (b,c) ranging over an ‘interesting’ part of the parameter plane (and for ρ fixed), the structure of the system is determined.

Summary of the results for the shallow lake system. Fig. 1 contains a lot of information on the shallow lake system. The parameter b characterises the physical properties of a lake, and may be seen as given. For such a b, there is the following sequence for increasing values of c (see also Fig. 4): unique equilibrium, which is a polluted lake; several equilibria, but the optimal usage leads to a polluted state; several equilibria, and the final state of the lake depends on its initial state; several equilibria, and the optimal usage leads to a clean lake; unique equilibrium, which is a clean lake. As can be seen from the figure, not all of these have to occur, depending on the value of b.

Moreover, recall the remark on the dynamic game above: if n agents use the lake, the system can be seen as a single-agent system, with the value of c replaced by c/n. The figure then shows—as was to be expected intuitively—that the final condition of the lake worsens as n increases, and that if n is large enough, the optimal solution will inevitably lead to a polluted lake.

Finally, note that the existence of a Skiba point has an important consequence for a social planner: since at a Skiba initial state the two optimal solutions are equivalent in economic terms, the decision which one to choose has to be based on other criteria. For instance, in the case of a shallow lake, environmental protection agencies might argue that it is morally good to opt for a clean environment.

General theory. The second part of the article gives a general analysis of the occurrence of Skiba points for systems with one phase variable and one control, for which the effect of the control variable has a definite direction. That is, if

$$\text{x}\text{̇}\text{=f(x,u)}$$

is the state equation, the effect of the control is said to have a definite direction if f is continuous and if

$$\text{∂f}\text{∂u}\text{≠0}$$

for any x and u. It is shown that for such a system the parameter region where Skiba points occur is bounded by heteroclinic bifurcation curves. Moreover, it is shown that if the state–co-state system has for ρ=0 a cusp bifurcation point, then for small ρ>0, there exist Skiba points in the system.