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Dynamic Games and Applications

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16.09.2023

Rational Noncooperative Strategic Exploitation of Species in a Predator–Prey Ecosystem with Random Disturbances

verfasst von: Christos Koulovatianos

Erschienen in: Dynamic Games and Applications

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Abstract

Ecological instability caused by pollution, climate change, or by exogenous distortions in the food chain of biological organisms may increase the average natural death rate of certain species, or it may increase the variance of their natural death rate, or both. Here, rational noncooperative strategic harvesting in a predator–prey ecosystem that is subject to exogenous environmental disturbances is studied through an example that delivers analytical solutions. When players exploit only one of two interacting species, then in symmetric Markovian–Nash equilibrium: (i) the ‘tragedy of the commons’ holds and (ii) when exogenous factors increase and/or make more volatile the natural geometric death rate of the species under exploitation (of the non-harvested species) each player’s harvesting rate increases (decreases) and the commons problem is intensified (mitigated).

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Fußnoten
1
In textbook analysis (see, for example, Clark [14]), this setting is often seen as a starting point, as a benchmark, because a path-independent rational-expectations equilibrium elucidates how harvesting incentives depend on model fundamentals (tastes of exploiting agents and natural laws of species reproduction).
 
2
I am not aware of any empirical study backing up this claim. Assuming that a more volatile environment, e.g., higher variation in temperature, or more frequent extreme events are likely to compromise the survival rate of species, is possibly a reasonable conjecture.
 
3
For a discussion of these mathematical difficulties and previous work concerning both multispecies models and stochastic models of natural-resource management, see the survey by Munro and Scott ([36], pp. 646–657) and also Clark ([14], Ch. 10 and 11).
 
4
For example, a famous predator–prey model is the “Lotka–Volterra” model. An economic application of this model can be found in Brander and Taylor [10].
 
5
An introduction to these difficulties, with a special discussion about the undesirable properties of linear-quadratic models, can be found in Mirman [35]. Difficulties concerning the characterization of the mechanics of models with general functional forms have motivated research through parametric examples, as is, e.g., the example of strategic harvesting under uncertainty in Antoniadou et al. [1] and the example of oligopolistic resource exploitation by Koulovatianos and Mirman [23].
 
6
Another example of a predator–prey ecosystem that helps in characterizing steady states and their stability properties are described in Ströbele and Wacker [41]. Yet, Ströbele and Wacker [41] do not study exploitation games but a single controller who exploits species that interact with other biological populations. Moreover, Chiarella et al. [13] focus on the open-loop equilibrium concept of multi-state-variable differential games of common-property resource exploitation that can accommodate predator–prey interactions; they provide conditions under which an open-loop equilibrium is socially optimal, a case where commonality of access does not matter.
 
7
Notice that if the prey is being caught at a high rate, this does not necessarily mean that the predator’s ‘per-capita’ consumption is also high. It can be that the stock of the predators is ‘too high’ relative to its prey, so as to observe a high death rate for the prey (the prey is caught more often), but also a low life expectancy for the predator (the predator’s ‘per-capita’ consumption is low).
 
8
To verify this linear form, notice that ẋ\(\left( t\right) =\left( 1-\theta \right) x\left( t\right) ^{-\theta }\dot{x}\left( t\right) \), ẏ\(\left( t\right) =\left( 1-\theta \right) y\left( t\right) ^{-\theta }\dot{y}\left( t\right) \) and multiply (1) and (2) accordingly.
 
9
So, unlike the fashionable, among ecologists, models of complex population dynamics, such as the “Lotka–Volterra” predator–prey model, the present example deals with an intrinsically stable ecosystem. Without a disposition to ignore a significant empirical literature on long-term population cycles observed for certain species (see, for example, Steele and Henderson [40]), the simplification of global stability of the ecosystem helps in pursuing a theoretical investigation of the complex game-theoretic issues regarding strategic exploitation. Nevertheless, not all ecosystems are surprisingly complex so as to cast the property of strong natural stability empirically irrelevant.
 
10
In order to overcome this limitation, of discrete-time models with logarithmic momentary utility, Antoniadou et al. [1] have introduced a constant-relative-risk aversion (CRRA) utility function, tied with a constant-elasticity of substitution reproduction function in a single-species analysis.
 
11
I am grateful to an anonymous referee for this clarification.
 
12
Other papers, such as, e.g., Benhabib and Radner [9], Dutta and Sundaram [18], and Sorger [38], deal with the issue of equilibrium multiplicity. This issue is outside the scope of this paper.
 
13
Notice that, unlike a discrete-time analysis, admissibility of an interior exploitation strategy of the form \(C^{x,i}\left( x\left( t\right) ,y\left( t\right) \right) =\omega _{x}x\left( t\right) \) does not require proving that \(\omega _{x}\in \left( 0,1\right) \). In continuous time, admissibility and interiority require only that \(\omega _{x}>0\) and then conditions guaranteeing the strict positivity of the population stocks \(\left( x\left( t\right) ,y\left( t\right) \right) \). I am grateful to a referee for asking me to clarify this point.
 
14
Notice that \(Det\left( M_{x}\right) >0\), while \(tr\left( M_{x}\right) <0\).
 
15
\(\left\lfloor a\right\rfloor \) denotes the largest integer smaller than or equal to a.
 
16
In Antoniadou et al. [1] it is found that the tragedy of the commons always holds in the case of exploitation of a single population without interaction with other species for the Markovian-Nash equilibrium in linear symmetric strategies for all games that have such an equilibrium.
 
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Metadaten
Titel
Rational Noncooperative Strategic Exploitation of Species in a Predator–Prey Ecosystem with Random Disturbances
verfasst von
Christos Koulovatianos
Publikationsdatum
16.09.2023
Verlag
Springer US
Erschienen in
Dynamic Games and Applications
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-023-00527-6