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The effects of random and seasonal environmental fluctuations on optimal harvesting and stocking

Alexandru Hening, Ky Q. Tran, Sergiu C. Ungureanu

Abstract

We analyze the harvesting and stocking of a population that is affected by random and seasonal environmental fluctuations. The main novelty comes from having three layers of environmental fluctuations. The first layer is due to the environment switching at random times between different environmental states. This is similar to having sudden environmental changes or catastrophes. The second layer is due to seasonal variation, where there is a significant change in the dynamics between seasons. Finally, the third layer is due to the constant presence of environmental stochasticity -- between the seasonal or random regime switches, the species is affected by fluctuations which can be modelled by white noise. This framework is more realistic because it can capture both significant random and deterministic environmental shifts as well as small and frequent fluctuations in abiotic factors. Our framework also allows for the price or cost of harvesting to change deterministically and stochastically, something that is more realistic from an economic point of view. The combined effects of seasonal and random fluctuations make it impossible to find the optimal harvesting-stocking strategy analytically. We get around this roadblock by developing rigorous numerical approximations and proving that they converge to the optimal harvesting-stocking strategy. We apply our methods to multiple population models and explore how prices, or costs, and environmental fluctuations influence the optimal harvesting-stocking strategy. We show that in many situations the optimal way of harvesting and stocking is not of threshold type.

The effects of random and seasonal environmental fluctuations on optimal harvesting and stocking

Abstract

We analyze the harvesting and stocking of a population that is affected by random and seasonal environmental fluctuations. The main novelty comes from having three layers of environmental fluctuations. The first layer is due to the environment switching at random times between different environmental states. This is similar to having sudden environmental changes or catastrophes. The second layer is due to seasonal variation, where there is a significant change in the dynamics between seasons. Finally, the third layer is due to the constant presence of environmental stochasticity -- between the seasonal or random regime switches, the species is affected by fluctuations which can be modelled by white noise. This framework is more realistic because it can capture both significant random and deterministic environmental shifts as well as small and frequent fluctuations in abiotic factors. Our framework also allows for the price or cost of harvesting to change deterministically and stochastically, something that is more realistic from an economic point of view. The combined effects of seasonal and random fluctuations make it impossible to find the optimal harvesting-stocking strategy analytically. We get around this roadblock by developing rigorous numerical approximations and proving that they converge to the optimal harvesting-stocking strategy. We apply our methods to multiple population models and explore how prices, or costs, and environmental fluctuations influence the optimal harvesting-stocking strategy. We show that in many situations the optimal way of harvesting and stocking is not of threshold type.

Paper Structure

This paper contains 31 sections, 7 theorems, 113 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Numerical Scheme
  4. Extensions
  5. Variable effort harvesting-stocking strategies
  6. Constructing the approximation scheme.
  7. Uncertain price functions
  8. Constructing the approximation scheme.
  9. The combined effects of seasonality and Markovian switching
  10. Constructing the approximation scheme.
  11. Numerical Examples
  12. Introducing switching in a Verhulst-Pearl system
  13. Analysis of the effect of the control cost
  14. Large and small transition rates with switching
  15. Switching in the cost and price functions
  16. ...and 16 more sections

Key Result

Theorem 2.4

Suppose Assumptions a:1 holds. Then for any $(x, \alpha) \in {\mathbb R}_+\times {\mathcal{M}}$, $V^h(x, \alpha)\to V(x, \alpha)$ as $h\to 0$. Thus, for sufficiently small $h$, a near-optimal harvesting-stocking strategy of the controlled Markov chain $\{(X^h_n, \alpha^h_n)\}$ is also a near-optimal

Figures (12)

  • Figure 1: Value function (left) and optimal harvesting-stocking rate (right) for a model with switching affecting $\mu(\alpha) = 4 - \alpha$, compared with a baseline model with no switching and $\mu(\alpha) = 2.5.$
  • Figure 2: Value function (left) and optimal harvesting-stocking rate (right) for a model with switching affecting $\mu(\alpha) = 4 - \alpha$, and convex cost $C(x,\alpha,u) = u^2/2$, compared with a baseline model with no switching, $\mu(\alpha) = 2.5$, and the same cost function.
  • Figure 3: The left and right panels show the same value function graph, but on different vertical scales. The model is discussed in Sections \ref{['sec:cost_num']}, \ref{['sec:transition_rates']}. The green solid curves in both panels corresponds to a model without switching and $\mu = 2.5$, or to a model with $\mu = 4 - \alpha$ and infinitely large switching rates. The blue long-dashed curves in the left panel show models with no switching and $\mu = 3$, $\mu=2$ respectively, or equivalently a model with $\mu = 4-\alpha$ and infinitely slow switching rates. The red dashed curves in the left panel are for a model with $\mu = 4-\alpha$ and $q_{12} = q_{21} = 0.01$. The black dotted curves in both panels correspond to the same model, but $q_{12}=q_{21}=0.1$. The teal dot-dashed curves in the right panel correspond to $q_{12}=q_{21}=1$, and the orange dashed curves in the right panel to $q_{12}=q_{21}=10$.
  • Figure 4: The left and right panels show the optimal harvesting functions for the model in Section \ref{['sec:cost_num']}, for two price functions, $p(u) = 1 - u/4$ on the left, and $p(u) = (1+u/3)^{-1}$ on the right. There is no dependency on $\alpha$ here, and $\mu = 2.5$. The other parameters are as before.
  • Figure 5: The left and right panels show the value function and optimal harvesting rate for a model with variable effort harvesting, and switching that affects the population growth rate. $\mu(\alpha) = 2 - \alpha/2$, $\kappa(\alpha) = 1/2$, and the cost function is $u^2$. The other parameters are as in Section \ref{['sec:intro_lv']}. For simplicity, we show only the value function and the harvesting strategy for $\alpha=1$. The other state has similar values and controls.
  • ...and 7 more figures

Theorems & Definitions (17)

  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Conjecture 4.1
  • Remark 4.2
  • Lemma A.1
  • Definition C.1
  • ...and 7 more