Bifurcation Curve Diagrams for a Diffusive Generalized Logistic Problem with Minkowski Curvature Operator and Constant-Yield Harvesting

TL;DR

This work analyzes positive solutions to a one-dimensional diffusive problem with Minkowski curvature and constant harvesting, modeled by $-((u')/\sqrt{1-(u')^2})' = \lambda g(u) - \mu$ on $(-L,L)$. Using a time-map framework, the authors prove that the bifurcation curves on the $(\lambda,\|u\|_\infty)$-plane and the $(\mu,\|u\|_\infty)$-plane are $C$-shaped (and reversed-$C$-shaped, respectively) with a single turning point, and they characterize the exact multiplicity of positive solutions via a bifurcation set on the $(\mu,\lambda)$-plane. The results are anchored by an auxiliary time-map $T_{\mu,\lambda}(\alpha)$ and detailed lemmas, with a lengthy appendix providing the rigorous proof of a key technical lemma. This extends classical logistic-bifurcation insights to a curvature-constrained diffusion setting and offers precise criteria for solution multiplicity under harvesting.

Abstract

This paper investigates the bifurcation diagrams of positive solutions for a one-dimensional diffusive generalized logistic boundary-value problem with the Minkowski curvature operator and constant yield harvesting. We prove that the corresponding bifurcation curves on both the (lambda, sup-norm of u)-plane and the (mu, sup-norm of u)-plane are C-shaped. Furthermore, by characterizing the bifurcation set on the (mu, lambda)-plane, we determine the exact multiplicity of positive solutions.

Figures (7)

• Figure 1: Graphs of $S_{\mu }$. $S_{\mu }$ is monotone increasing for $\mu =0$, and $\subset$-shaped for $\mu >0$. (i) $g^{\prime }(0^{+})=\infty$. (ii) $g^{\prime }(0^{+})\in (0,\infty )$.
• Figure 2: Graphs of $\Sigma _{\lambda }$. $\Sigma _{\lambda }$ is reversed $\subset$-shaped for $\lambda >\kappa$. (i) $\kappa <\lambda \leq 2\kappa .$ (ii) $\lambda >2\kappa$.
• Figure 3: The bifurcation set $B_{\Gamma }$. (i) $g^{\prime }(0^{+})=\infty$. (ii) $g^{\prime }(0^{+})\in (0,\infty )$.
• Figure 4: The projection of the curves $C_{\Gamma }=C_{1}\cup C_{2}$ onto the first quadrant of the $\left( \mu ,\lambda \right)$-plane. (i) $g^{\prime }(0^{+})=\infty$. (ii) $g^{\prime }(0^{+})\in (0,\infty )$.
• Figure 5: Graphs of $T_{0,\lambda }$ and $T_{\mu ,\lambda }$. (i) $\eta >0$ (i.e. $g^{\prime }(0^{+})\in (0,\infty )$). (ii) $\eta =0$ (i.e. $g^{\prime }(0^{+})=\infty$).

Theorems & Definitions (17)

• Lemma 1: Hung
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4: See Figure \ref{['fig3']}
• Example 1
• Lemma 2
• Lemma 3: See Figure \ref{['fig8']}
• Lemma 4