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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.06179 |
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| _version_ | 1866915483994816512 |
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| author | de la Rosa, Rafael Medina, Elena |
| author_facet | de la Rosa, Rafael Medina, Elena |
| contents | In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation $u_t\,=\,D\,(u^{ν-1}\,u_x)_x\,+\,a\,u^μ$, with $μ,ν>0$, accompanied by hostile boundary conditions and examine two families of one-parametric initial distributions, including homogeneous distributions. The model was previously studied by Colombo and Anteneodo (2018). They determined appropriate habitat sizes $l$ for the survival of a population, whose individuals are initially placed homogeneously within the full habitat domain with a total initial population $n_0$. We show that the survival condition can be naturally formulated in terms of the parameter $Q:=\frac{a}{D}l^{-μ+ν+2}n_0^{μ-ν}$. Indeed, there exists a critical value $Q_c$ determined by $μ$, $ν$ and the initial distribution parameter such that the survival condition can always be written as $Q\geq Q_c$. Notably, from this point of view, one can derive a condition for $Q$ that holds universally for our model under conditional persistence ($μ\geqν$). It applies, in particular, to the case $μ=ν+2$, which was not addressed in the previously mentioned work. Nevertheless, in this case $Q=\frac{a}{D}n_0^2$, therefore survival depends solely on the total population, not on the habitat size. We apply a finite-difference scheme to estimate $Q_c$. Conversely, given a population whose evolution is determined by $μ$, $ν$, $l$, $n_0$, and the growth and diffusion coefficients $a$ and $D$ (and consequently the value of $Q$) we use the numerical algorithm to estimate the initial distribution to ensure population survival. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06179 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The role of the initial distribution in population survival within a bounded habitat de la Rosa, Rafael Medina, Elena Analysis of PDEs Numerical Analysis In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation $u_t\,=\,D\,(u^{ν-1}\,u_x)_x\,+\,a\,u^μ$, with $μ,ν>0$, accompanied by hostile boundary conditions and examine two families of one-parametric initial distributions, including homogeneous distributions. The model was previously studied by Colombo and Anteneodo (2018). They determined appropriate habitat sizes $l$ for the survival of a population, whose individuals are initially placed homogeneously within the full habitat domain with a total initial population $n_0$. We show that the survival condition can be naturally formulated in terms of the parameter $Q:=\frac{a}{D}l^{-μ+ν+2}n_0^{μ-ν}$. Indeed, there exists a critical value $Q_c$ determined by $μ$, $ν$ and the initial distribution parameter such that the survival condition can always be written as $Q\geq Q_c$. Notably, from this point of view, one can derive a condition for $Q$ that holds universally for our model under conditional persistence ($μ\geqν$). It applies, in particular, to the case $μ=ν+2$, which was not addressed in the previously mentioned work. Nevertheless, in this case $Q=\frac{a}{D}n_0^2$, therefore survival depends solely on the total population, not on the habitat size. We apply a finite-difference scheme to estimate $Q_c$. Conversely, given a population whose evolution is determined by $μ$, $ν$, $l$, $n_0$, and the growth and diffusion coefficients $a$ and $D$ (and consequently the value of $Q$) we use the numerical algorithm to estimate the initial distribution to ensure population survival. |
| title | The role of the initial distribution in population survival within a bounded habitat |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2509.06179 |