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- Open Access
An exact bifurcation diagram for a reaction–diffusion equation arising in population dynamics
- Jerome Goddard II^{1}Email authorView ORCID ID profile,
- Quinn A. Morris^{2},
- Stephen B. Robinson^{3} and
- Ratnasingham Shivaji^{4}
- Received: 21 March 2018
- Accepted: 1 November 2018
- Published: 15 November 2018
Abstract
Keywords
- Mathematical biology
- Reaction–diffusion model
- Bifurcation
- Stability
MSC
- 35J25
1 Introduction
Habitat fragmentation creates landscape-level spatial heterogeneity which influences the population dynamics of the resident species. Of particular interest, fragmentation often leads to declines in abundance of the species as the fragmented landscape becomes more susceptible to edge effects between the remnant habitat patches and the lower quality human-modified “matrix” surrounding these focal patches [1–3]. Studies of movement behavior in response to different habitat edge conditions clearly demonstrate that the composition of the matrix can influence emigration rates, patterns of movement, and within-patch distributions of a species (e.g., [4–6]). Movement behavior has been shown to be very species-specific [7], even in the same fragmented habitat.
Even though the task of connecting the wealth of empirical information available about individual movement and mortality in response to matrix composition to predictions about patch-level persistence is indeed formidable, the reaction–diffusion framework and its underlying random walk models have seen some success in addressing this connection [8]. The reaction–diffusion framework’s major strength is that the model’s dynamics can be analyzed mathematically, providing important patch-level predictions of population persistence, leading to its wide adoption by ecologists ([9, 10], and [11]). This framework is also ideally suited to handle fragmentation and edge-mediated effects as the partial differential equation(s) involved require explicit definition of edge behavior via boundary conditions ([10] and [2]).
In [12], the authors formalize a framework to facilitate the connection between small-scale movement and patch-level predictions of persistence through a mechanistic model based on reaction–diffusion equations. The model is capable of incorporating essential information about edge-mediated effects such as patch preference, movement behavior (e.g., emigration rates, patterns of movement, and within-patch distributions), and matrix-induced mortality at the patch/matrix interface. The authors then mathematically analyze the model’s predictions of persistence with a general logistic-type growth term. In particular, the focus of [12] is to provide bounds on demographic attributes and patch size in order for the model to predict persistence of a species in a given patch based on assumptions on the patch/matrix interface and to explore their sensitivity to demographic attributes both in the patch and the matrix, as well as patch size and geometry. The purpose of this present work is to provide an exact description of the bifurcation curve of positive steady states to their model when the growth term is logistic. In the following subsections, we will briefly summarize the modeling framework and boundary condition derivation given in [12] and present our main results. We provide the proof of these results in Sect. 2. In Sect. 3, for the case \(n=1\), we provide an alternative proof of our results in the case \(\Omega=(0,1)\) via a quadrature method and discuss the evolution of the bifurcation curves as a model parameter varies. We discuss biological implications of our results in Sect. 4. Finally, in Appendix 1, we provide the derivation of the boundary condition focused on in this paper, and in Appendix 2, we provide results on certain eigenvalue problems that we employ in the proof of our main result.
1.1 Modeling framework
Listing of interface scenarios with descriptions and selected references from [12]
Scenario name | Scenario description | κ | References |
---|---|---|---|
Continuous density | Organisms move between the patch and the matrix with equal probability. Step sizes and movement probabilities are equal in the patch and the matrix. | 1 | [15] |
Type I Discontinuous density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in or leaving Ω different from 50%. Step sizes differ between the patch and the matrix, whereas movement probabilities are equal. | \(\frac{\alpha}{1 - \alpha} \sqrt{\frac {D_{0}}{D}}\) | |
Type II Discontinuous density (DD) | Organisms modify their movement behavior at the patch/matrix interface and would have a probability α of remaining in or leaving Ω different from 50%. Step sizes are equal between the patch and the matrix but movement probabilities are different. | \(\frac{\alpha}{1 - \alpha} \frac {D_{0}}{D}\) | |
Type III Discontinuous density (DD) | Organisms remain in Ω with probability α different from 50%. Movement probabilities and step sizes are the same between the patch and the matrix. | \(\frac{\alpha}{1 - \alpha}\) |
Interface scenario | γ-value |
---|---|
Continuous density | \(\sqrt{\frac{S_{0} D_{0}}{r D}}\) |
Type I DD | \(\frac{1 - \alpha}{\alpha} \frac{\sqrt{S_{0}}}{\sqrt {r}}\) |
Type II DD | \(\frac{1 - \alpha}{\alpha} \frac{\sqrt{S_{0} D}}{\sqrt {r D_{0}}}\) |
Type III DD | \(\frac{1 - \alpha}{\alpha} \sqrt{\frac{S_{0} D_{0}}{r D}}\) |
1.2 Statement of the main result
Theorem 1
- (a)
If \(\lambda> \lambda_{1} (\gamma)\), then the trivial solution of (4) is unstable and there exists a unique positive solution \(v_{\lambda}\) to (4) which is globally asymptotically stable. Furthermore, \(\|v_{\lambda}\|_{\infty}\to0^{+}\) as \(\lambda\to\lambda_{1} (\gamma)^{+}\) and \(\|v_{\lambda}\|_{\infty}\to1\) as \(\lambda\to\infty\);
- (b)
If \(0 < \lambda\leq\lambda_{1} (\gamma)\), then the trivial solution of (4) is globally asymptotically stable and there is no positive solution to (4).
2 Proof of Theorem 1
In this section, we provide a proof of our main results given in Theorem 1.
Proof
Lemma 1
Note that by Lemma 5 in Appendix 2, we have that \(\operatorname{sign}(\delta_{1}(\lambda, \gamma)) = \operatorname{sign}(\sigma_{1}(\lambda , \gamma))\). Thus, it suffices to only consider the sign of \(\sigma _{1}(\lambda, \gamma)\).
Since \(\frac{\partial\overline{\psi}}{\partial\eta} < 0\) in \(\Omega_{0}\), clearly \(\overline{\psi}_{\lambda}\) is a subsolution to (4) for \(\lambda\gg1\), and since \(w=1\) is a supersolution to (4) for \(\lambda> 0\), we must have \(v_{\lambda}\in [ \overline{\psi}_{\lambda}, 1 ]\) and hence \(\|v_{\lambda}\|_{\infty}\to1^{-}\) as \(\lambda\to\infty\).
Now, to show the stability properties of \(v_{\lambda}\), recall that we have \(\psi= m_{1} \phi\) is a strict subsolution for all \(m_{1} \in (0, -\frac{\sigma_{1}}{\lambda} )\) and \(Z \equiv M\) is a strict supersolution for all \(M > 1\). This implies that \(\phi< v_{\lambda}< Z\) and ϕ can be made arbitrarily small and Z can be made arbitrarily large. This fact combined with a result such as Theorem 6.7 of Chap. 5 in [19] immediately shows that \(v_{\lambda}\) is globally asymptotically stable, proving (a).
3 One-dimensional problem
4 Biological implications of our results
These model results give important predictions on population persistence at the patch level based solely on demographic parameters, e.g., patch diffusion rate and intrinsic growth rate, as well as matrix diffusion rate and death rate. We note that our analysis covers all four of the interface scenarios listed in Table 1. Although the exact definition of γ depends upon the interface scenario given (see Table 2), the qualitative behavior of the persistence of the organism is the same as γ varies. Notice that the principal eigenvalue \(\sigma_{1}(\lambda, \gamma)\) of (6) plays a crucial role in determining the dynamics of the model. In fact, it represents the fastest possible growth rate for the linear growth model corresponding to (3) (see [3] or [10]).
As indicated in Theorem 1 (and the proof therein), when \(\lambda\leq\lambda_{1}(\gamma)\) we have that \(\sigma_{1}(\lambda, \gamma) \geq0\), and the only nonnegative steady state of (3) is the trivial one, \(u \equiv0\). In this case, the model predicts extinction for any nonnegative initial population density profile. In fact, loses due to mortality in the matrix outpace the reproductive rate in the patch. Thus, the theoretical organism cannot colonize the patch, and any remnant population in the patch will become extinct. However, when \(\lambda> \lambda_{1}(\gamma)\), we have that \(\sigma_{1}(\lambda, \gamma) < 0\) and (3) admits a unique steady state that is positive in Ω̅ such that all positive initial population density profiles will propagate to this steady state over time. In this case, the global nature of the stability of the positive steady state gives a fairly strong notion of persistence of the species. The patch is large enough in this case to shield a sufficient proportion of the population from mortality induced by the hostile matrix. This prediction leads to a formula for minimum patch size of the population given as \(\ell^{*}(\gamma) = \sqrt{\frac {D}{r} \lambda_{1}(\gamma)}\). Note that this formula can be numerically estimated and depends upon parameters in the patch (diffusion rate and intrinsic growth rate), parameters in the matrix via γ (diffusion rate and death rate), and the geometry of the patch \(\Omega_{0}\).
This notion of a minimum patch size agrees with the well-known notion of a minimum core area (in the case of \(n = 2\)) requirement. Note that \(\lambda_{1}(\gamma)\) can be viewed as a quantification of the loss of the population due to interactions with the hostile matrix where γ encapsulates parameters regarding the hostile matrix. Also, it is easy to see that \(\lambda_{1}(\gamma) \to\lambda_{1}^{D}\) as \(\gamma\to\infty\), and this reveals an important model prediction of the existence of a maximum possible effect of population loss due to the hostile matrix. Patches with a lethal matrix can still guarantee a prediction of persistence as long as the patch size is larger than \(\sqrt{\frac{D}{r} \lambda_{1}^{D}}\), where the maximum effect of the lethal matrix on the population is quantified in \(\lambda_{1}^{D}\). This minimum patch size approaches infinity if either (1) the patch diffusion rate is arbitrarily large, since a large diffusion rate ensures that a very high proportion of the population will encounter loss at the patch/matrix interface, or (2) the intrinsic growth rate is arbitrarily small, which for a fixed patch diffusion rate will imply that the population is not able to recover the loss associated with interaction with hostile matrix.
Declarations
Acknowledgements
The authors would like to thank to the anonymous reviewers whose suggestions greatly improved this manuscript.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This work was supported in part by the National Science Foundation via grant DMS-1516560 for the first author and grant DMS-1516519 for the last author.
Authors’ contributions
The authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Ries, L., Fletcher, R.J. Jr., Battin, J., Sisk, T.D.: Ecological responses to habitat edges: mechanisms, models, and variability explained. Annu. Rev. Ecol. Evol. Syst. 35(1), 491–522 (2004) View ArticleGoogle Scholar
- Fagan, W.F., Cantrell, R.S., Cosner, C.: How habitat edges change species interactions. Am. Nat. 153(2), 165–182 (1999) View ArticleGoogle Scholar
- Cantrell, R.S., Cosner, C., Fagan, W.F.: Competitive reversals inside ecological reserves: the role of external habitat degradation. J. Math. Biol. 37(6), 491–533 (1998) MathSciNetView ArticleGoogle Scholar
- Tscharntke, T., Steffan-Dewenter, I., Kruess, A., Thies, C.: Contribution of small habitat fragments to conservation of insect communities of grassland–cropland landscapes. Ecol. Appl. 12(2), 354–363 (2002) Google Scholar
- Schooley, R.L., Wiens, J.A.: Movements of cactus bugs: patch transfers, matrix resistance, and edge permeability. Landsc. Ecol. 19(7), 801–810 (2004) View ArticleGoogle Scholar
- Haynes, K.J., Cronin, J.T.: Interpatch movement and edge effects: the role of behavioral responses to the landscape matrix. Oikos 113(1), 43–54 (2006) View ArticleGoogle Scholar
- Reeve, J.D., Cronin, J.T.: Edge behaviour in a minute parasitic wasp. J. Anim. Ecol. 79(2), 483–490 (2010) View ArticleGoogle Scholar
- Maciel, G.A., Lutscher, F.: How individual movement response to habitat edges affects population persistence and spatial spread. Am. Nat. 182(1), 42–52 (2013) View ArticleGoogle Scholar
- Turchin, P.: Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer, Sunderland (1998) Google Scholar
- Cantrell, R.S., Cosner, C.: Spatial Ecology Via Reaction–Diffusion Equations. Mathematical and Computational Biology, p. 411. Wiley, Chichester (2003) MATHGoogle Scholar
- Holmes, E.E., Lewis, M.A., Banks, R.R.V.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75(1), 17–29 (1994) View ArticleGoogle Scholar
- Cronin, J.T., Goddard, J. II, Shivaji, R.: Effects of patch matrix and individual movement response on population persistence at the patch-level. Preprint (2017) Google Scholar
- Maciel, G., Lutscher, F.: How individual movement response to habitat edges affects population persistence and spatial spread. Am. Nat. 182(1), 42–52 (2013) View ArticleGoogle Scholar
- Okubo, A.: Diffusion and Ecological Problems: Mathematical Models. Biomathematics, vol. 10. Springer, Berlin (1980) MATHGoogle Scholar
- Ludwig, D., Aronson, D.G., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979) MathSciNetView ArticleGoogle Scholar
- Ovaskainen, O., Cornell, S.J.: Biased movement at a boundary and conditional occupancy times for diffusion processes. J. Appl. Probab. 40(3), 557–580 (2003) MathSciNetView ArticleGoogle Scholar
- Cantrell, R.S., Cosner, C.: Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor. Popul. Biol. 55(2), 189–207 (1999) View ArticleGoogle Scholar
- Cantrell, R.S., Cosner, C.: Density dependent behavior at habitat boundaries and the Allee effect. Bull. Math. Biol. 69, 2339–2360 (2007) MathSciNetView ArticleGoogle Scholar
- Pao, C.V.: Nonlinear Parabolic and Elliptic Equations, p. 777. Plenum, New York (1992) MATHGoogle Scholar
- Goddard, J. II, Shivaji, R.: Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions. Proc. R. Soc. Edinb. A 147, 1019–1040 (2017) MathSciNetView ArticleGoogle Scholar
- Inkmann, F.: Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions. Indiana Univ. Math. J. 31(2), 213–221 (1982) MathSciNetView ArticleGoogle Scholar
- Laetsch, T.: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J. 20, 1–13 (1970/71) MathSciNetView ArticleGoogle Scholar
- Anuradha, V., Maya, C., Shivaji, R.: Positive solutions for a class of nonlinear boundary value problems with Neumann–Robin boundary conditions. J. Math. Anal. Appl. 236(1), 94–124 (1999) MathSciNetView ArticleGoogle Scholar
- Goddard, J. II, Lee, E.K., Shivaji, R.: Population models with nonlinear boundary conditions. Electron. J. Differ. Equ. Conf. 19, 135–149 (2010) MathSciNetMATHGoogle Scholar
- Goddard, J. II, Morris, Q., Son, B., Shivaji, R.: Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions. Electron. J. Differ. Equ. 2018, 26 (2018) MathSciNetView ArticleGoogle Scholar
- Goddard, J. II, Price, J., Shivaji, R.: Analysis of steady states for classes of reaction–diffusion equations with U-shaped density dependent dispersal on the boundary (2017, in press) Google Scholar
- Miciano, A.R., Shivaji, R.: Multiple positive solutions for a class of semipositone Neumann two point boundary value problems. J. Math. Anal. Appl. 178(1), 102–115 (1993) MathSciNetView ArticleGoogle Scholar
- Rivas, M.A., Robinson, S.: Eigencurves for linear elliptic equations. European Journal ESAIM (to appear) Google Scholar