# Spatially Distributed Differential Game Theoretic Model of Fisheries

^{*}

## Abstract

**:**

## 1. Introduction and Related Work

## 2. The Initial Model of the Ecosystem of a Shallow Body of Water with Consideration of Fishing

## 3. The Averaged Model and the QRS Method in Simulation Modeling

- (a)
- for $\forall \text{\hspace{0.17em}}{(v,\text{\hspace{0.17em}}w)}^{(i)},{(v,w)}^{(j)}\in QRS$ we have $|{J}_{0}{}^{(i)}-{J}_{0}{}^{(j)}|/{J}_{0}^{\mathrm{max}}>\Delta $;
- (b)
- for $\forall {(v,w)}^{(l)}\notin QRS$ $\exists {(v,u)}^{(j)}\in QRS:$ $|{J}_{0}{}^{(l)}-{J}_{0}{}^{(j)}|/{J}_{0}^{\mathrm{max}}\le \Delta $.

## 4. A Discrete Version of the Model

## 5. Computer Simulation on the Base of the QRS Method

- The form and values of all input functions and parameters for the model in question are defined.
- One of potential QRS is chosen as a current supervisor’s strategy.
- The set of Nash equilibria is built by the complete enumeration on the QRS set of all agents under the fixed supervisor’s strategy. The problems in Equations (5) and (6) are solved numerically.
- The pair of strategies (a current supervisor’s strategy and the set of best responses of the agents to it) is compared with the current best pair of strategies for the supervisor (from the point of view of Equation (11)). The current strategy becomes the optimal one if necessary.
- If the number of feasible scenarios for the supervisor are not exhausted, then her new strategy is chosen and the return to step 3 of the algorithm is implemented.

- All input functions and parameters for the model in question are defined.
- The strategies of punishment of the agents by the supervisor are introduced as ${({v}^{(m)})}^{p}(t)={\left\{{({v}_{ij}^{(m)})}^{p}\right\}}_{i,j=1}^{M(K)}$$${\left\{{({v}_{ij}^{(m)})}^{p}\right\}}_{i,j=1}^{M(K)}=\mathrm{arg}\underset{{\left\{{v}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)}}{\mathrm{min}}\underset{{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)}}{\mathrm{max}}{J}_{m}({\left\{{s}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)},{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)},P(t,x)),$$$${L}_{m}=\underset{{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)}}{\mathrm{max}}{J}_{m}({\left\{{s}_{ij}^{(p)}\right\}}_{i,j=1}^{M(K)},{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)},P(t,x)),$$Denote ${\left\{{({w}_{ij}^{(m)})}^{p}\right\}}_{i,j=1}^{M(K)}=\mathrm{arg}\underset{{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)}}{\mathrm{max}}{J}_{m}({\left\{{({v}_{ij}^{(m)})}^{p}\right\}}_{i,j=1}^{M(K)},{\left\{{w}_{ij}^{(m)}\right\}}_{i,j=1}^{M(K)},P(t,x))$.
- The best supervisor’s strategy (from the point of view of the payoff functional (11)) is found by the enumeration of her scenarios from the QRS set under the conditions$${J}_{m}(s,w,P)>{L}_{m};m=1,2,\dots ,n.$$This strategy is called the reward strategy, and the last inequality provides an advantage of the reward strategy in comparison with the punishment strategy for each agent. Denote the solution of this optimization problem by $\left({\left\{{({v}_{ij}^{(m)})}^{R}\right\}}_{i,j=1}^{M(K)},{\left\{{({w}_{ij}^{(m)})}^{R}\right\}}_{i,j=1}^{M(K)}\right)$.
- If at the previous step it is impossible to find the reward strategy, i.e., there is at least one agent (with index ${m}_{0}$) for whom the inequality ${J}_{m}(s,w,P)>{L}_{m}$ is not satisfied on the QRS set then the supervisor punishes this agent, and the inverse Stackelberg equilibrium has the form $\left({\left\{{({v}_{ij}^{({m}_{0})})}^{P}\right\}}_{i,j=1}^{M(K)},{\left\{{({w}_{ij}^{({m}_{0})})}^{P}\right\}}_{i,j=1}^{M(K)}\right)$.
- The equilibrium strategies for other agents are the following:$$\left({\left\{{({v}_{ij}^{(m)})}^{R}\right\}}_{i,j=1}^{M(K)},{\left\{{({w}_{ij}^{(m)})}^{R}\right\}}_{i,j=1}^{M(K)}\right)\text{\hspace{0.17em}},m=1,2,\dots ,n;\text{\hspace{0.17em}}m\ne {m}_{0}.$$

## 6. Numerical Results

**Example.**Suppose that $M=K=2;\text{}N=1$; ${P}_{00}=0.5$ th.u.; ${\gamma}_{1}=0.3\text{\hspace{0.17em}}\mathrm{th}.\mathrm{u}.;{\gamma}_{2}=0.3$; $D=0.1$ m

^{2}; $T=365$ d.; $L=100$ m.; ${C}_{1}=0.16$; ${C}_{2}=1$ (th.u.)

^{−1}, ${C}_{3}=0.2$; $E=0.16\text{\hspace{0.17em}}\mathrm{r}.$/(d.m.); $\rho =0.01$; $H=600$ r./(d.m. (th.u.)²); ${\gamma}_{3}=0.001\text{\hspace{0.17em}}\mathrm{th}.\mathrm{u}.$; ${P}^{\ast}=0.6$ th.u.; $a=b=2000$ r./(d.m.th.u.). In the case of compulsion ${s}_{ij}\equiv 0.6,$ and in the case of impulsion ${q}_{ij}\equiv 0\text{\hspace{0.17em}};i,j=1,2;$ r.—roubles; d.—days; th.u.—thousand units of fish biomass; m—meter.

- -
- when the input model parameters vary in a wide range the system compatibility index does not vary essentially. Therefore, a hierarchical control is required to ensure the sustainable management;
- -
- an increase of the price of unit of fish biomass results in the increase of the agent’s payoff. The supervisor’s payoff does not change. The SCI is close to one. The system becomes more compatible;
- -
- an increase of the capture cost results in the decrease of the agent’s payoff. The supervisor’s payoff does not decrease. As in the previous case, the system becomes more compatible;
- -
- an increase of the penalty coefficient leads to the decrease of the supervisor’s payoff. The system under compulsion becomes less compatible, and the role of supervisor in the system increases;
- -
- the best information structure for the supervisor is an inverse Stackelberg game under compulsion;
- -
- in contrast, the best information structure for the agents is a Stackelberg game under impulsion;
- -
- for most of the considered input data the SCI is closer to one in Stackelberg games under impulsion.

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Clark, C.W. The Worldwide Crisis in Fisheries: Economic Models and Human Behavior; Cambridge University Press: Cambridge, UK, 2006; 270p. [Google Scholar]
- Arnason, R. Fisheries management and operations research. Eur. J. Oper. Res.
**2009**, 193, 741–751. [Google Scholar] [CrossRef] - Doyen, L.; Bene, C. Sustainability of fisheries through marine reserves a robust modeling analysis. J. Environ. Manag.
**2003**, 69, 1–13. [Google Scholar] [CrossRef] - Anderies, J.M.; Rodriguez, A.A.; Janssen, M.A.; Cifdaloz, O. Panaceas, uncertainty, and the robust control framework in sustainability science. Proc. Natl. Acad. Sci. USA
**2007**, 104, 15194–15199. [Google Scholar] [CrossRef] [Green Version] - Aubin, J.-P. Viability Theory; Springer: Berlin, Germany, 1991; 342p. [Google Scholar]
- Bailey, M.; Rashid Sumaila, U.; Lindroos, M. Application of game theory to fisheries over three decades. Fish. Res.
**2010**, 102, 1–8. [Google Scholar] [CrossRef] - Kaitala, V.; Lindroos, M. Game theoretic application to fisheries. In Handbook of Operations Research on Natural Resources; Springer: Berlin, Germany, 2007; pp. 201–216. [Google Scholar]
- Munro, G.R. The optimal management of transboundary renewable resources. Can. J. Econ.
**1979**, 12, 355–376. [Google Scholar] [CrossRef] - Levhari, D.; Mirman, L.J. The great fish war: An example using a dynamic Cournot-Nash solution. Bell J. Econ.
**1980**, 11, 322–334. [Google Scholar] [CrossRef] - Sumaila, U.R. A review of game-theoretic models of fishing. Mar. Pollut.
**1999**, 23, 1–10. [Google Scholar] [CrossRef] - Yi, S.-S. Stable coalition structures with externalities. Games Econ. Behav.
**1997**, 20, 201–237. [Google Scholar] [CrossRef] - Pintassilgo, P. A coalition approach to the management of high seas fisheries in the presence of externalities. Nat. Res. Model.
**2003**, 16, 175–196. [Google Scholar] [CrossRef] - Kronbak, L.G.; Lindroos, M. Sharing rules and stability in coalition games with externalities. Mar. Res. Econ.
**2007**, 22, 137–154. [Google Scholar] [CrossRef] - Wirl, F. Tragedy of the Commons in a Stochastic Game of a Stock Externality. J. Public Econ. Theory
**2008**, 10, 99–124. [Google Scholar] [CrossRef] - Wang, W.-K.; Ewald, C.-O. A stochastic differential Fishery game for a two species fish population with ecological interaction. J. Econ. Dyn. Control
**2010**, 34, 844–857. [Google Scholar] [CrossRef] - Clarke, F.H.; Munro, G.R. Coastal states, distant water fishing nations and extended jurisdiction: A principal-agent analysis. Nat. Res. Model.
**1987**, 2, 81–107. [Google Scholar] [CrossRef] - Mesterton-Gibbons, M. Game-theoretic resource modeling. Nat. Res. Model.
**1993**, 7, 93–147. [Google Scholar] [CrossRef] - Clarke, F.H.; Munro, G. Coastal states and distant water fishing nations: Conflicting views of the future. Nat. Res. Model.
**1991**, 5, 345–369. [Google Scholar] [CrossRef] - Jensen, F.; Vestergaard, N. A principal-agent analysis of fisheries. J. Inst. Theor. Econ.
**2002**, 158, 276–285. [Google Scholar] [CrossRef] - Bailey, M.; Sumaila, U.R. Destructive fishing in Raja Ampat, Indonesia: An applied principal-agent analysis. Fish. Cent. Res. Rep.
**2008**, 16, 142–169. [Google Scholar] - Rettieva, A.N. A discrete-time bioresource management problem with asymmetric players. Autom. Remote Control
**2014**, 75, 1665–1676. [Google Scholar] [CrossRef] - Rettieva, A.N. A bioresource management problem with different planning horizons. Autom. Remote Control
**2015**, 76, 919–934. [Google Scholar] [CrossRef] - Abakumov, A.I.; Il’in, O.I.; Ivanko, N.S. Game problems of harvesting in a biological community. Autom. Remote Control
**2016**, 77, 697–707. [Google Scholar] [CrossRef] - Sukhinov, A.I.; Nikitina, A.V.; Chistyakov, A.E. Modeling the Scenario of Biological Rehabilitation of Azov Sea. Mat. Modelir.
**2012**, 24, 3–21. [Google Scholar] - Sukhinov, A.I.; Chistyakov, A.E.; Alekseenko, E.V. A Numerical Implementation of the Three Dimensional Hydrodynamical Model for Shallow Reservoirs with a Supercomputer. Mat. Modelir.
**2011**, 23, 3–21. [Google Scholar] - Sukhinov, A.I.; Chistyakov, A.E. Adapting a Modified Alternating Triangle Iterative Method for Solving Grid Equations with Non-Self-Adjoint Operator. Mater. Modelir.
**2012**, 24, 3–20. [Google Scholar] - Sukhinov, A.I.; Chistyakov, A.E.; Shishenya, A.V.; Timofeeva, E.F. Predictive Modeling of Coastal Hydrophysical Processes in Multiple-Processor Systems Based on Explicit Schemes. Math. Models Comput. Simul.
**2018**, 10, 648–658. [Google Scholar] [CrossRef] - Nikitina, A.V.; Sukhinov, A.I.; Ugolnitskii, G.A.; Usov, A.B.; Chistyakov, A.E.; Puchkin, M.V.; Semenov, I.S. Optimal control of sustainable development in the biological rehabilitation of the Azov Sea. Math. Models Comput. Simul.
**2017**, 9, 101–107. [Google Scholar] [CrossRef] - Sukhinov, A.I.; Chistyakov, A.E.; Ugolnitskii, G.A.; Usov, A.B.; Nikitina, A.V.; Puchkin, M.V.; Semenov, I.S. Game-Theoretic Regulations for Control Mechanisms of Sustainable Development for Shallow Water Ecosystems. Autom. Remote Control
**2017**, 78, 1059–1071. [Google Scholar] [CrossRef] - Basar, T.; Olsder, G.J. Dynamic Non-Cooperative Game Theory; Society for Industrial & Applied Mathematics (SIAM): Philadelphia, PA, USA, 1999; 536p. [Google Scholar]
- Olsder, G.J. Phenomena in inverse Stackelberg games. Рart 2: Dynamic problems. J. Optim. Theory Appl.
**2009**, 143, 601–618. [Google Scholar] [CrossRef] - Gorelov, M.A.; Kononenko, A.F. Dynamic models of conflicts. III. Hierarchical games. Autom. Remote Control
**2015**, 76, 264–277. [Google Scholar] [CrossRef] - Beven, K. Spatially Distributed Modeling: Conceptual Approach to Runoff Prediction. In Recent Advances in the Modeling of Hydrological Systems; Springer Science: Berlin, Germany, 1991; pp. 373–387. [Google Scholar]
- Ougolnitsky, G. Sustainable Management; Nova Science Publishers: Hauppauge, NY, USA, 2011; 287p. [Google Scholar]
- Ougolnitsky, G.A. Sustainable Management as a Key to Sustainable Development. In Sustainable Development: Processes, Challenges and Prospects; Reyes, D., Ed.; Nova Science Publishers: Hauppauge, NY, USA, 2015; pp. 87–128. [Google Scholar]
- Ougolnitsky, G.A. Differential Games in Environmental Management. In Environmental Management: Past, Present and Future; Wright, E., Ed.; Nova Science Publishers: Hauppauge, NY, USA, 2017; pp. 1–52. [Google Scholar]
- Ugol’nitskii, G.A.; Usov, A.B. A study of differential models for hierarchical control systems via their discretization. Autom. Remote Control
**2013**, 74, 252–263. [Google Scholar] [CrossRef] - Ugol’nitskii, G.A.; Usov, A.B. Equilibria in models of hierarchically organized dynamic systems with regard to sustainable development conditions. Autom. Remote Control
**2014**, 75, 1055–1068. [Google Scholar] [CrossRef] - Ougolnitsky, G.A.; Usov, A.B. Computer Simulation as a Method of Solution of Differential Games. In Computer Simulations: Advances in Research and Applications; Pfeffer, M.D., Bachmaier, E., Eds.; Nova Science Publishers: Hauppauge, NY, USA, 2018; pp. 63–106. [Google Scholar]
- Murray, J.D. Lectures on Nonlinear Differential Equation Models in Biology; Clarendon Press: Oxford, UK, 1977; 370p. [Google Scholar]
- Hamming, R.W. Numerical Methods for Scientists and Engineers; Dover Publications: Hauppauge, NY, USA, 1987; 752p. [Google Scholar]

**Table 1.**Examination of the condition (а) for the built initial QRS set for the example input data in the case of compulsion.

$\mathbf{Strategies}\text{}{(\mathit{q},\mathit{w})}^{(\mathit{j})}$ | ${\mathit{J}}_{0}{}^{(\mathit{j})}(\mathit{q},\mathit{w})$ | $\underset{\begin{array}{l}1\le \mathit{k}\le 2401\\ \mathit{k}\ne \mathit{j}\end{array}}{\mathbf{min}}\frac{\left|{\mathit{J}}_{0}{}^{(\mathit{k})}-{\mathit{J}}_{0}{}^{(\mathit{j})}\right|}{{\mathit{J}}_{0}{}^{\mathbf{min}}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{q}}_{\mathbf{00}}$ | ${\mathit{q}}_{\mathbf{01}}$ | ${\mathit{q}}_{\mathbf{10}}$ | ${\mathit{q}}_{\mathbf{11}}$ | ${\mathit{w}}_{\mathbf{00}}$ | ${\mathit{w}}_{\mathbf{01}}$ | ${\mathit{w}}_{\mathbf{10}}$ | ${\mathit{w}}_{\mathbf{11}}$ | ||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1,288,689.1 | 0.29 |

0.5 | 0 | 0.5 | 0 | 0 | 0 | 0.5 | 0 | 1,611,411.3 | 0.006 |

0.5 | 0 | 0.5 | 0 | 0.25 | 0 | 0.25 | 0 | 730,788.0 | 0.23 |

0.5 | 0 | 0 | 0 | 0.5 | 0 | 0 | 0 | 794,262.9 | 0.6 |

0 | 0.5 | 0.5 | 0 | 0 | 0.5 | 0.25 | 0 | 667,493.1 | 0.005 |

1 | 0 | 0 | 0 | 0.5 | 0 | 0 | 0 | 749,681.1 | 0.25 |

0 | 0 | 1 | 1 | 0 | 0 | 0.5 | 0.5 | 1,100,280.2 | 0.01 |

0 | 0 | 0 | 1 | 0 | 0 | 0 | 0.5 | 1,192,904.6 | 0.01 |

1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 628,276.1 | 0.11 |

1 | 0.5 | 1 | 0.5 | 0 | 0 | 0.5 | 0 | 2,097,953.3 | 0.9 |

0 | 0 | 0.5 | 0.5 | 0 | 0 | 0.5 | 0.5 | 1,100,145.3 | 0.002 |

0 | 0.5 | 0.5 | 0.5 | 0 | 0.25 | 0.5 | 0.25 | 636,455.1 | 0.11 |

1 | 0 | 1 | 0 | 0 | 0 | 0.5 | 0 | 1,611,897.2 | 0.006 |

0.5 | 0 | 0 | 1 | 0.25 | 0 | 0 | 0.5 | 713,713.5 | 0.23 |

0 | 1 | 0.5 | 0.5 | 0 | 0.5 | 0.25 | 0.5 | 589,451.5 | 0.52 |

0 | 1 | 0.5 | 0 | 0 | 0.5 | 0.25 | 0 | 667,911.3 | 0.005 |

0.5 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 74,353.9 | 0.003 |

0.5 | 0.5 | 0.5 | 0.5 | 0 | 0 | 0 | 0 | 2,260,802.0 | 0.01 |

0.5 | 0.5 | 0.5 | 0.5 | 0.25 | 0.25 | 0.25 | 0.25 | 177,846.8 | 1 |

0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 73,935.8 | 0.001 |

0.5 | 0.5 | 1 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 74,070.7 | 0.001 |

1 | 1 | 1 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 74,906.9 | 0.007 |

1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 2,261,773.2 | 0.01 |

1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1,172,296.6 | 0.26 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,192,054.2 | 0.01 |

**Table 2.**Examination of the condition (b) for the example qualitatively representative scenarios (QRS) set under compulsion.

Strategies ${(\mathit{q},\mathit{w})}^{(\mathit{j})}\notin \mathit{Q}\mathit{R}\mathit{S}$ | «Close» Strategies ${(\mathit{q},\mathit{w})}^{(\mathit{k})}\in \mathit{Q}\mathit{R}\mathit{S}$ | $\frac{\left|{\mathit{J}}_{0}^{(\mathit{k})}-{\mathit{J}}_{0}^{(\mathit{j})}\right|}{{\mathit{J}}_{0}^{\mathbf{min}}}$ | ||
---|---|---|---|---|

${\mathit{q}}^{(\mathit{j})}$ | ${\mathit{w}}^{(\mathit{j})}$ | ${\mathit{q}}^{(\mathit{k})}$ | ${\mathit{w}}^{(\mathit{k})}$ | |

$\left(\begin{array}{cc}0.1& 0.2\\ 0.1& 0.1\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0.1& 0.1\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ | 0.025 |

$\left(\begin{array}{cc}0.3& 0.1\\ 0.2& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}0.3& 0.1\\ 0.1& 0.1\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0\\ 0& 0\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0\\ 0& 0\end{array}\right)$ | 0.015 |

$\left(\begin{array}{cc}0.1& 0.8\\ 0.4& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.1& 0.4\\ 0.2& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0& 1\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0& 0.5\\ 0.25& 0.5\end{array}\right)$ | 0.007 |

$\left(\begin{array}{cc}0.1& 0.6\\ 0.6& 0\end{array}\right)$ | $\left(\begin{array}{cc}0.1& 0.4\\ 0.3& 0\end{array}\right)$ | $\left(\begin{array}{cc}0& 0.5\\ 0.5& 0\end{array}\right)$ | $\left(\begin{array}{cc}0& 0.5\\ 0.25& 0\end{array}\right)$ | 0.002 |

$\left(\begin{array}{cc}0.1& 0\\ 0.4& 0.6\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0.4& 0.6\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0.5& 0.5\end{array}\right)$ | 0.018 |

$\left(\begin{array}{cc}0.6& 0.4\\ 0.5& 0.7\end{array}\right)$ | $\left(\begin{array}{cc}0.3& 0.4\\ 0.4& 0.6\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0.5\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0.5\\ 0.5& 0.5\end{array}\right)$ | 0.006 |

$\left(\begin{array}{cc}0.9& 0.4\\ 0.8& 0.6\end{array}\right)$ | $\left(\begin{array}{cc}0.8& 0.4\\ 0.8& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}1& 0.5\\ 1& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}1& 0.5\\ 1& 0.5\end{array}\right)$ | 0.023 |

$\left(\begin{array}{cc}0.4& 0.1\\ 0.2& 0.9\end{array}\right)$ | $\left(\begin{array}{cc}0.2& 0\\ 0.1& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}0.25& 0\\ 0& 0.5\end{array}\right)$ | 0.014 |

$\left(\begin{array}{cc}0.9& 0.1\\ 0.9& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}0.1& 0.2\\ 0.4& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 1& 0\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0.5& 0\end{array}\right)$ | 0.007 |

$\left(\begin{array}{cc}0.4& 0.4\\ 0.6& 0.3\end{array}\right)$ | $\left(\begin{array}{cc}0.2& 0.1\\ 0.1& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0.5\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ | 0.021 |

$\left(\begin{array}{cc}0.9& 0.8\\ 0.8& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}0.9& 0.8\\ 0.8& 0.1\end{array}\right)$ | $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$ | $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$ | 0.037 |

$\left(\begin{array}{cc}0.1& 0.4\\ 0.6& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.1& 0.2\\ 0.6& 0.2\end{array}\right)$ | $\left(\begin{array}{cc}0& 0.5\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0& 0.25\\ 0.5& 0.25\end{array}\right)$ | 0.018 |

$\left(\begin{array}{cc}0.6& 0.9\\ 0.6& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.4& 0.6\\ 0.4& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 1\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0.5\\ 0.5& 0.5\end{array}\right)$ | 0.014 |

$\left(\begin{array}{cc}0.9& 0.8\\ 0.8& 0.9\end{array}\right)$ | $\left(\begin{array}{cc}0.9& 0.8\\ 0.8& 0.8\end{array}\right)$ | $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ | $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ | 0.019 |

$\left(\begin{array}{cc}0.6& 0.4\\ 0.4& 0.4\end{array}\right)$ | $\left(\begin{array}{cc}0.2& 0.3\\ 0.3& 0.3\end{array}\right)$ | $\left(\begin{array}{cc}0.5& 0.5\\ 0.5& 0.5\end{array}\right)$ | $\left(\begin{array}{cc}0.25& 0.25\\ 0.25& 0.25\end{array}\right)$ | 0.007 |

**Table 3.**Dependence of the system compatibility index (SCI) on the price of fish biomass unit for the example input data.

$\mathit{a}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_1}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_2}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_1}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_2}$ |
---|---|---|---|---|

10,000 | 1.33 | 1.34 | 1.15 | 1.25 |

2000 | 1.36 | 1.37 | 1.24 | 1.36 |

100 | 1.37 | 1.39 | 1.36 | 1.4 |

$\mathit{b}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_1}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_2}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_1}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_2}$ |
---|---|---|---|---|

10,000 | 1.33 | 1.35 | 1.16 | 1.25 |

2000 | 1.36 | 1.37 | 1.24 | 1.36 |

100 | 1.38 | 1.4 | 1.37 | 1.42 |

$\mathit{H}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_1}$ | ${\mathit{\kappa}}_{\mathit{c}\mathit{o}\mathit{m}\_2}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_1}$ | ${\mathit{\kappa}}_{\mathit{i}\mathit{m}\mathit{p}\_2}$ |
---|---|---|---|---|

100 | 1.25 | 1.26 | 1.18 | 1.23 |

600 | 1.36 | 1.37 | 1.24 | 1.36 |

2000 | 1.5 | 1.54 | 1.32 | 1.52 |

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Ougolnitsky, G.; Usov, A.
Spatially Distributed Differential Game Theoretic Model of Fisheries. *Mathematics* **2019**, *7*, 732.
https://doi.org/10.3390/math7080732

**AMA Style**

Ougolnitsky G, Usov A.
Spatially Distributed Differential Game Theoretic Model of Fisheries. *Mathematics*. 2019; 7(8):732.
https://doi.org/10.3390/math7080732

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Ougolnitsky, Guennady, and Anatoly Usov.
2019. "Spatially Distributed Differential Game Theoretic Model of Fisheries" *Mathematics* 7, no. 8: 732.
https://doi.org/10.3390/math7080732