Optimal harvesting for a logistic model with grazing

Mohan Mallick; Ardra A; Sarath Sasi

doi:10.3934/eect.2024066

Article Contents

2025, Volume 14, Issue 3: 511-529. Doi: 10.3934/eect.2024066

This issue Previous Article Stabilization with decay estimate of infinite dimensional second order systems by bilinear control damping Next Article Stabilization and parameter estimation for 1-D wave equation with variable coefficients and uncertain harmonic disturbances

Optimal harvesting for a logistic model with grazing

1.
VNIT Nagpur, 440010, India
2.
IIT Palakkad, Kerala, 678623, India

^*Corresponding author: Sarath Sasi
^*Corresponding author: Sarath Sasi

Received: January 2024

Revised: October 2024

Early access: November 2024

Published: June 2025

The second author was supported by the Department of Science and Technology INSPIRE Fellowship of the Government of India.

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Abstract

We consider semi-linear elliptic equations of the following form:

$ \begin{equation*} \left\{ \begin{aligned} -\Delta u & = \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u] = :\lambda f_h(u) &&{\rm{in}} \; \Omega, \\ \frac{\partial u}{\partial \eta}&+qu = 0 &&{\rm{on}} \; \partial\Omega, \end{aligned} \right. \end{equation*} $

where, $ h\in U = \{h\in L^2(\Omega): 0\leq h(x)\leq H\}. $ We prove the existence and uniqueness of the positive solution for large $ \lambda. $ Further, we establish the existence of an optimal control $ h\in U $ that maximizes the functional $ J(h) = \int_{\Omega}h(x)u_h(x)\; {\rm{d}}x-\int_{\Omega}(B_1+B_2 h(x))h(x)\; {\rm{d}}x $ over $ U $, where $ u_h $ is the unique positive solution of the above problem associated with $ h $, $ B_1>0 $ is the cost per unit effort when the level of effort is low and $ B_2>0 $ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.

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Mathematics Subject Classification: Primary: 49J20, 49K20, 92D25, 92D40; Secondary: 35J05, 35P05.

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Figure 1. Graph of $ f_{h_{\beta}} $ with $ \beta = 0.1, $ $ K = 17, $ and $ c = 1.5 $

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Figure 2. Graphs illustrating the arguments given in Proposition 2.2

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References

[1]	R. A. Adams and J. J. F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam), Elsevier, 140 (2003).
[2]	N. Barlow, Harvesting models for resource-limited populations, New Zealand Journal of Ecology, (1987), 129-133.
[3]	A. Cañada, J. L. Gámez and J. A. Montero, Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type, SIAM Journal on Control and Optimization, 36 (1998), 1171-1189. doi: 10.1137/S0363012995293323.
[4]	R. S. Cantrell and C. Cosner, Density dependent behavior at habitat boundaries and the Allee effect, Bulletin of Mathematical Biology, 69 (2007), 2339-2360. doi: 10.1007/s11538-007-9222-0.
[5]	C. W. Clark, Bioeconomic Modelling and Fisheries Management, New York: Wiley, 1985.
[6]	W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling, 22 (2009), 173-211. doi: 10.1111/j.1939-7445.2008.00033.x.
[7]	M. Mallick, et al., Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions, Communications on Pure & Applied Analysis, (2021), 705-726. doi: 10.3934/cpaa.2021195.
[8]	G. Fragnelli, D. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Advanced Nonlinear Studies, 16 (2016), 603-622. doi: 10.1515/ans-2016-0010.
[9]	L. Gerber, et al., Population models for marine reserve design: A retrospective and prospective synthesis, Ecological Applications, 13 (2003), 47-64.
[10]	H. S. Gordon, The economic theory of a common-property resource: The fishery, Journal of Political Economy, 62 (1954). doi: 10.1086/257497.
[11]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing Inc, 1985.
[12]	S. Heikkilä and V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 181 (1994).
[13]	J. Jiang and J. Shi, Bistability dynamics in some structured ecological models, Spatial Ecology, Chapman & Hall/CRC, Mathematical and Computational Biology Series, (2010), 33-62. doi: 10.1201/9781420059861.ch3.
[14]	E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, Journal of Mathematical Analysis and Applications, 381 (2011), 732-741. doi: 10.1016/j.jmaa.2011.03.048.
[15]	E. Lee, S. Sasi and R. Shivaji, An ecological model with a $\Sigma$-shaped bifurcation curve, Nonlinear Analysis: Real World Applications, 13 (2012), 634-642. doi: 10.1016/j.nonrwa.2011.07.054.
[16]	A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, Journal of Mathematical Analysis and Applications, 173 (1993), 603-619. doi: 10.1006/jmaa.1993.1091.
[17]	R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
[18]	M. G. Neubert, Marine reserves and optimal harvesting, Ecology Letters, 6 (2003), 843-849. doi: 10.1046/j.1461-0248.2003.00493.x.
[19]	I. Noy-Meir, Stability of grazing systems an application of predator-prey graphs, The Journal of Ecology, 63 (1975), 459-482. doi: 10.2307/2258730.
[20]	J. Sanchirico and J. Wilen, A bioeconomic model of marine reserve creation, Journal of Environmental Economics and Management, 42 (2001), 257-276. doi: 10.1006/jeem.2000.1162.
[21]	J. Sanchirico and J. Wilen, Optimal spatial management of renewable resources: Matching policy scope to ecosystem scale, Journal of Environmental Economics and Management, 50 (2005), 23-46. doi: 10.1016/j.jeem.2004.11.001.
[22]	J. Steele and E. Henderson, Modelling long term fluctuations in fish stocks, Science, 224 (1984), 985-987. doi: 10.1126/science.224.4652.985.
[23]	E. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems, Ecology, 86 (2005), 1797-1807. doi: 10.1890/04-0550.