A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch
We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation: \begin{equation*} \begin{split} -&u^{\prime \prime}= \begin{cases} \lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\i...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2020-03-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7934 |
| _version_ | 1827951134384324608 |
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| author | Nalin Fonseka Jonathan Machado Ratnasingham Shivaji |
| author_facet | Nalin Fonseka Jonathan Machado Ratnasingham Shivaji |
| author_sort | Nalin Fonseka |
| collection | DOAJ |
| description | We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation:
\begin{equation*}
\begin{split}
-&u^{\prime \prime}=
\begin{cases}
\lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\in [L,1-L] ,\\
\lambda \left[u-\dfrac{1}{K}u^2\right];& x\in (0,L)\cup(1-L,1),
\end{cases}
\\
-&u^{\prime}(0) + \sqrt{\lambda}\gamma u(0) = 0,\\
&u^{\prime}(1) + \sqrt{\lambda}\gamma u(1) = 0,\\
\end{split}
\end{equation*}
where $\lambda$, $c$, $K$, and $\gamma$ are positive parameters and the parameter $L\in(0,\frac{1}{2})$. The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat $\Omega_0=(0,1)$, where grazing (type of predation) is occurring on the subregion $[L,1-L]$. In this model, $u$ is the population density and $c$ is the maximum grazing rate. $\lambda$ is a parameter which influences the equation as well as the boundary conditions, and $\gamma$ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet ($\gamma \longrightarrow \infty$). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when $\gamma$ is finite, and their evolutions as $\gamma$ and $L$ vary. |
| first_indexed | 2024-04-09T13:37:02Z |
| format | Article |
| id | doaj.art-2f059a616595405c8a604783ab1d514a |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:02Z |
| publishDate | 2020-03-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-2f059a616595405c8a604783ab1d514a2023-05-09T07:53:10ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752020-03-0120201711110.14232/ejqtde.2020.1.177934A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patchNalin Fonseka0Jonathan Machado1Ratnasingham Shivaji2Ph.D Student in Computational Mathematics, Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USAUndergraduate Mathematics Major, Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USADept of Mathematics & Statistics, 116 Petty, 317 College Ave., UNCG, Greensboro, NC 27412, USAWe will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation: \begin{equation*} \begin{split} -&u^{\prime \prime}= \begin{cases} \lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\in [L,1-L] ,\\ \lambda \left[u-\dfrac{1}{K}u^2\right];& x\in (0,L)\cup(1-L,1), \end{cases} \\ -&u^{\prime}(0) + \sqrt{\lambda}\gamma u(0) = 0,\\ &u^{\prime}(1) + \sqrt{\lambda}\gamma u(1) = 0,\\ \end{split} \end{equation*} where $\lambda$, $c$, $K$, and $\gamma$ are positive parameters and the parameter $L\in(0,\frac{1}{2})$. The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat $\Omega_0=(0,1)$, where grazing (type of predation) is occurring on the subregion $[L,1-L]$. In this model, $u$ is the population density and $c$ is the maximum grazing rate. $\lambda$ is a parameter which influences the equation as well as the boundary conditions, and $\gamma$ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet ($\gamma \longrightarrow \infty$). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when $\gamma$ is finite, and their evolutions as $\gamma$ and $L$ vary.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7934differential equationsboundary value problemslogistic growthexterior matrix hostilityinterior grazingpositive solutions |
| spellingShingle | Nalin Fonseka Jonathan Machado Ratnasingham Shivaji A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch Electronic Journal of Qualitative Theory of Differential Equations differential equations boundary value problems logistic growth exterior matrix hostility interior grazing positive solutions |
| title | A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| title_full | A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| title_fullStr | A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| title_full_unstemmed | A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| title_short | A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| title_sort | study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch |
| topic | differential equations boundary value problems logistic growth exterior matrix hostility interior grazing positive solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7934 |
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