# Optimal Harvest Problem for Fish Population—Structural Stabilization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Model for Populations: Solution’s Properties

#### 2.1. Model Description

#### 2.2. Optimization System

- (a)
- Set $D\subset {R}^{n}$ is a closed bounded set with smooth boundary $\partial D$, and set U of the possible control functions is closed and convex. The set D has a positive Lebesgue measure, and set U is not empty.
- (b)
- The control function $u\left(t,x\right)$ is piecewise continuous. All other functions are continuous and have the smoothness of a required degree.
- (c)
- Problem (1), (2) has single solution y(t, x) for each possible control function u, and function y locally satisfies the Lipschitz condition on variable u, i.e., $\parallel \delta y\parallel \le M\cdot \parallel \delta u\parallel $ for small increments δu of function u and corresponding increments δy of function y for some M > 0. We denote as $\parallel \xb7\parallel $ a norm in space $C\left(T\times D\right)$ of continuous functions.

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

## 3. Modelling of Population

#### 3.1. Homogeneous Population

- The functions $h\left(y,u\right)$ and c$\left(u\right)$ are monotone increasing and linear functions at the variable u. The solution exists in this case for a confined set $U=\left[0,{u}_{m}\right]$. The optimal solution has the control function $u\left(t\right)$ as a piecewise constant function in the “bang-bang” form. For $c\left(u\right)\equiv 0$ this assumes the form$$u\left(t\right)=\left\{\begin{array}{c}0fort\in \left[0,{t}_{1}\right]\\ {u}_{a}fort\in \left[{t}_{1},{t}_{2}\right]\\ {u}_{m}fort\in \left[{t}_{2},{t}_{m}\right]\end{array}\right.$$
- The harvesting function $h\left(y,u\right)$ is a linear function at the variable u. The expense function $c\left(u\right)$ has form $c\left(u\right)={c}_{0}{u}^{\alpha}$ with some constants ${c}_{0}>0,\alpha 1$. The optimal control function $u\left(t\right)$ is continuous.

#### 3.2. Size Structure—Fish Population Characteristics

#### 3.3. Model without Diffusion

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

#### 3.4. Unstable Environments Characteristics

#### 3.5. Parameters of Fishery

#### 3.6. Numerical Method

## 4. Results

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Joffe, A.D.; Tikhomirov, V.M. Theory of Extremal Problems; Studies in Mathematics and Its Applications Series; North-Holland Pub. Co.: Amsterdam, The Netherlands; New York, NY, USA, 1979; Volume 6. [Google Scholar]
- Berkovitz, L.D.; Medhin, L.G. Nonlinear Optimal Control Theory; CRC Press: London, UK; New York, NY, USA, 2013. [Google Scholar]
- Neubert, M.G. Marine reserves and optimal harvesting. Ecol. Lett.
**2003**, 6, 843–849. [Google Scholar] [CrossRef] - Winemiller, R.K.O. Life history strategies, population regulation, and implications for fisheries management. Can. J. Fish. Aquat. Sci.
**2005**, 62, 872–885. [Google Scholar] [CrossRef] - Perissi, I.; Bardi, U.; Asmar TEl Lavacchi, A. Dynamic patterns of overexploitation in fisheries. Ecol. Mod.
**2017**, 359, 285–292. [Google Scholar] [CrossRef] [PubMed] - Ricker, W.E. Computation and interpretation of biological statistics of fish populations. Bull. Fish. Res. Board Can.
**1975**, 191, 382. [Google Scholar] - Clark, C.W. Mathematical bioeconomics. In The Mathematics of Conservation, 3rd ed.; John Wiley and Sons Ltd.: Hoboken, NJ, USA, 2010. [Google Scholar]
- Fuller, E.C.; Samhouri, J.F.; Stoll, J.S.; Levin, S.A.; Watson, J.R. Characterizing fisheries connectivity in marine social–ecological systems. ICES J. Mar. Sci.
**2017**, 74, 2087–2096. [Google Scholar] [CrossRef] [Green Version] - Ricker, W.E. Handbook of computations for biological statistics of fish populations. Bull. Fish. Res. Board Can.
**1958**, 119, 300. [Google Scholar] - Paulik, G.J.; Hourston, A.S.; Larkin, P.A. Exploitation of multiple stocks by a common fishery. J. Fish. Res. Board Can.
**1958**, 24, 2527–2537. [Google Scholar] [CrossRef] - Hilborn, R. Optimal exploitation of multiple stocks by a common fishery: A new methodology. J. Fish. Res. Board Can.
**1976**, 33, 1–5. [Google Scholar] [CrossRef] [Green Version] - Matsuda, H.; Abrams, P.A. Maximal yields from multispecies fisheries systems: Rules for systems with multiple trophic levels. Ecol. Appl.
**2006**, 16, 225–237. [Google Scholar] [CrossRef] - Matsuda, H.; Makino, M.; Kotani, K. Optimal fishing policies that maximize sustainable ecosystem services. In Fisheries for Global Welfare and Environment, Proceedings of the 5th World Fisheries Congress 2008, Yokohama, Japan, 20–25 October 2008; Tsukamoto, K., Kawamura, T., Takeuchi, T., Beard, T.D., Kaiser, J.M.J., Eds.; Terrapub: Tokyo, Japan, 2008; pp. 359–369. [Google Scholar]
- Kar, T.K.; Ghosh, B. Impacts of maximum sustainable yield policy to prey-predator systems. Ecol. Model.
**2013**, 250, 134–142. [Google Scholar] [CrossRef] - Jacobsen, N.S.; Gislason, H.; Andersen, K.H. The consequences of balanced harvesting of fish communities. Proc. R. Soc. B Biol. Sci.
**2014**, 281, 20132701. [Google Scholar] [CrossRef] [Green Version] - Hilborn, R.; Walters, C.J. Role of Stock Assessment in Fisheries Management. In Quantitative Fisheries Stock Assessment; Springer: Boston, MA, USA, 1992. [Google Scholar] [CrossRef]
- Wolf, P. Recovery of the Pacific sardine and the California sardine fishery. CalCOFI Rep.
**1992**, 33, 76–86. [Google Scholar] - Watson, R.A.; Cheung, W.W.L.; Anticamara, J.A.; Sumaila, R.U.; Zeller, D.; Pauly, D. Global marine yield halved as fishing intensity redoubles. Fish Fish.
**2013**, 14, 493–503. [Google Scholar] [CrossRef] - Beverton, R.J.H.; Holt, S.J. On the dynamics of exploited fish populations. In Fishery Investigations; Series II; Ministry of Agriculture, Fisheries and Food: London, UK, 1957; Volume XIX. [Google Scholar]
- Flaaten, O. The optimal harvesting of a natural resource with seasonal growth. Can. J. Econ. Rev. Can. Econ.
**1983**, 16, 447–462. [Google Scholar] [CrossRef] - Johnson, M.; Sandell, J. Marine managed areas and fisheries. In Advances in Marine Biology Book 69, 1st ed.; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Abakumov, A.I.; Il’in, O.I.; Ivanko, N.S. Game problems of harvesting in a biological community. Autom. Remote Control Math. Game Theory Appl.
**2016**, 77, 697–707. [Google Scholar] [CrossRef] - Rodseth, T. Models for Multispecies Management; Physics-Verlag: Heidelberg, Germany; New York, NY, USA, 1998. [Google Scholar]
- Gurtin, M.E.; MacCamy, R.C. Nonlinear age-dependent population dynamics. Arch. Ration. Mech. Anal.
**1974**, 54, 281–300. [Google Scholar] [CrossRef] - Gurtin, M.E.; Murphy, L.F. On the optimal harvesting of persistent age-structured populations. J. Math. Biol.
**1981**, 13, 131–148. [Google Scholar] [CrossRef] - Brokate, M. Pontryagin’s principle for control problems in age-dependent population dynamics. J. Math. Biol.
**1985**, 23, 75–101. [Google Scholar] [CrossRef] - Murphy, L.F.; Smith, S.J. Optimal harvesting of an age-structured population. J. Math. Biol.
**1990**, 29, 77–90. [Google Scholar] [CrossRef] - Anita, S. Analysis and Control of Age-Dependent Population Dynamics; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar] [CrossRef]
- Ainseba, B.; Anita, S.; Langlais, L. Optimal control for a nonlinear age-structured; population dynamics model. Electron. J. Differ. Eq.
**2002**, 2002, 1–9. [Google Scholar] - Neverova, G.P.; Abakumov, A.I.; Yarovenko, I.P.; Frisman, E.Y. Mode change in the dynamics of exploited limited population with age structure. Nonlinear Dyn.
**2018**, 94, 827–844. [Google Scholar] [CrossRef] - Park, E.J.; Iannelli, M.; Kim, M.Y.; Anita, S. Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math.
**1998**, 58, 1648–1666. [Google Scholar] [CrossRef] - Kato, N. Optimal harvesting for nonlinear size-structured population dynamics. J. Math. Anal. Appl.
**2008**, 342, 1388–1398. [Google Scholar] [CrossRef] [Green Version] - Abia, L.M.; Angulo, O.; López-Marcos, J.C.; López-Marcos, M.A. Numerical schemes for a size-structured cell population model with equal fission. Math. Comput. Model.
**2009**, 50, 653–664. [Google Scholar] [CrossRef] - Hilborn, R.; Stokes, K. Defining overfished stocks: Have we lost the plot? Fisheries
**2010**, 35, 113–120. [Google Scholar] [CrossRef] - Schaefer, M.B. Some aspects of the dynamics of populations important to the management of commercial marine fisheries. J. Fish. Res. Board Can.
**1957**, 14, 669–681. [Google Scholar] [CrossRef] - Kot, M. Elements of Mathematical Ecology; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Parma, A. Optimal harvesting of fish populations with non-stationary stock recruitment relationships. Nat. Res. Mod.
**1990**, 4, 39–76. [Google Scholar] [CrossRef] - Maunder, M.N.; Thorson, J.T. Modeling temporal variation in recruitment in fisheries stock assessment: A review of theory and practice. Fish. Res.
**2019**, 217, 71–86. [Google Scholar] [CrossRef] - Carson, R.; Granger, W.J.C.; Jackson, J.B.C.; Schlenker, W. Fisheries management under cyclical population dynamics. Environ. Res. Econ.
**2006**, 42, 379. [Google Scholar] [CrossRef] [Green Version] - Martinet, V.; Peña-Torres, J.; De Lara, M.; Ramírez, C.H. Risk and Sustainability: Assessing fishery management strategies. Environ. Res. Econ.
**2016**, 64, 683. [Google Scholar] [CrossRef] - Cantrell, R.S.; Cosner, C. Spatial Ecology via Reaction-Diffusion Equations; John Wiley and Sons Ltd.: New York, NY, USA, 2003. [Google Scholar]
- Zorich, V.A. Mathematical Analysis II; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar] [CrossRef]
- Castilho, C.; Srinivasu, P.D.N. Bio-economics of a renewable resource in a seasonally varying environment. Math. Biosci.
**2007**, 205, 1–18. [Google Scholar] [CrossRef] [PubMed] - Hoar, W.S.; Randall, D.J.; Brett, J.R. Fish Physiology. Volume 8. Bioenergetics and Growth; Academic Press: New York, NY, USA; San Francisco, CA, USA; London, UK, 1979. [Google Scholar]
- Hart, P.J.B.; Reynolds, J.D. Handbook of Fish Biology and Fisheries, Volume 1. Fish Biology; Blackwell Publishing: Hoboken, NJ, USA, 2002. [Google Scholar]
- Svirezhev, Y.M.; Logofet, D.O. Stability of Biological Communities; Mir Publisher: Moscow, Russia, 1983. [Google Scholar]
- Moiseev, P.A. The Living Resources of the World Ocean; U.S. Department of the Interior: Washington, DC, USA, 1971.

**Figure 1.**The postulated population density $y\left(0,x\right)$ in equilibrium (

**left**) and the function ${m}_{1}\left(x\right)$ for natural mortality (

**right**) vs. size x.

**Figure 5.**The dynamics of catch volume ${{\displaystyle \int}}_{0}^{1}q\left(x\right)u\left(t,x\right)\psi \left(t\right)y\left(t,x\right)dx$.

**Figure 6.**The dynamics of the fishery part of the population ${{\displaystyle \int}}_{{x}_{0}}^{1}y\left(t,x\right)dx$.

Parameter | Description | Value |
---|---|---|

${y}_{0}$ | Initial population density of zero-sized individuals | 10^{2} |

${x}_{0}$ | Size at the entry into puberty | 0.35 |

$\alpha $ | Growth rate parameter | 0.83 |

$\mathsf{\gamma}$ | Parameter of the specific mortality function | 1.1 |

$k$ | Diffusion coefficient | 0.001 |

${\epsilon}_{\mathrm{max}}$ | Parameter of birthrate variation | 10 |

${\tau}_{m}$ | Maximum longevity | 25 |

${q}_{0}$$,{m}_{0}$ | Catchability coefficient and maximum natural mortality | 1.82 |

${p}_{0}$ | Parameter of catchability function | 4.0 |

${v}_{0}$ | Parameter of the speed of growth function | 0.24 |

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Abakumov, A.; Izrailsky, Y.
Optimal Harvest Problem for Fish Population—Structural Stabilization. *Mathematics* **2022**, *10*, 986.
https://doi.org/10.3390/math10060986

**AMA Style**

Abakumov A, Izrailsky Y.
Optimal Harvest Problem for Fish Population—Structural Stabilization. *Mathematics*. 2022; 10(6):986.
https://doi.org/10.3390/math10060986

**Chicago/Turabian Style**

Abakumov, Aleksandr, and Yuri Izrailsky.
2022. "Optimal Harvest Problem for Fish Population—Structural Stabilization" *Mathematics* 10, no. 6: 986.
https://doi.org/10.3390/math10060986