<p>In the last few decades, due to toxicants emitted by biological species through domestic wastage, industrialization, pesticides, burning garbage, etc., there have been rapidly abnormal changes in their function like productivity, change in shape, necrosis, etc. Most of the time, ecologist focused on their research on the existence or extinction of biological species but limited focus on abnormal deformation. Information is also supporting their theories. These abnormal changes are not instantaneous, i.e., they take their own time to show the results. Therefore, we are proposing and analyzing to investigate the impact of internally emitted toxicants by the species themselves in a four-compartment population model accompanied by the deformation delay(<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tau _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>) and in toxicant’s uptake delay(<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> through the biological population. Results show that when the deformation delay (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\tau _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>)is absent, the system is locally stable for any level of toxicant’s uptake delay (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\tau _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>) at coexisting equilibrium. However, when both delays occur during the procedure, the biological population system becomes unstable, i.e., the system exhibits Hopf bifurcation after reaching a critical value of deformation delay. Therefore, population density oscillates over time around the interior steady state. Additionally, to obtain the minimum cost to control the biological population’s deformation, we use optimal control theory through Pontryagin’s maximum principle. Finally, the numerically validated theoretical results.</p>

Study the Dynamics of a Deformed Population and Optimal Control due to Prolongation in Internally Emitted Toxicant’s Uptake

期刊论文
发表日期：

2025年8月7日

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Digvijai Singh,
Joydip Dhar,
Alok Kumar Agrawal

摘要

In the last few decades, due to toxicants emitted by biological species through domestic wastage, industrialization, pesticides, burning garbage, etc., there have been rapidly abnormal changes in their function like productivity, change in shape, necrosis, etc. Most of the time, ecologist focused on their research on the existence or extinction of biological species but limited focus on abnormal deformation. Information is also supporting their theories. These abnormal changes are not instantaneous, i.e., they take their own time to show the results. Therefore, we are proposing and analyzing to investigate the impact of internally emitted toxicants by the species themselves in a four-compartment population model accompanied by the deformation delay( $\tau _1$ $τ_{1}$ ) and in toxicant’s uptake delay( $\tau _2)$ $τ_{2})$ through the biological population. Results show that when the deformation delay ( $\tau _1$ $τ_{1}$ )is absent, the system is locally stable for any level of toxicant’s uptake delay ( $\tau _2$ $τ_{2}$ ) at coexisting equilibrium. However, when both delays occur during the procedure, the biological population system becomes unstable, i.e., the system exhibits Hopf bifurcation after reaching a critical value of deformation delay. Therefore, population density oscillates over time around the interior steady state. Additionally, to obtain the minimum cost to control the biological population’s deformation, we use optimal control theory through Pontryagin’s maximum principle. Finally, the numerically validated theoretical results.