A mathematical study of diffusive logistic equations with mixed type boundary conditions

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A mathematical study of diffusive logistic equations with mixed type boundary conditions

Institute of Mathematics, University of Tsukuba, Tsukuba 305–8571, Japan

Dedicated to the memory of Professor Rosella Mininni (1963–2020)

Received August 2021 Early access December 2021

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.

Keywords: Diffusive logistic equation, spatial heterogeneity, mixed type boundary condition, population dynamics, bifurcation theory, positive solution, super-subsolution method.

Mathematics Subject Classification: Primary: 35J65; Secondary: 35P30, 35J25, 92D25.

Citation: Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021166

References:

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References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2^nd edition, Pure and Applied Mathematics, Vol. 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar
[2]	G. A. Afrouzi and K. J. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proc. Amer. Math. Soc., 127 (1999), 125-130. doi: 10.1090/S0002-9939-99-04561-X. Google Scholar
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[4]	H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, North-Holland Math. Studies, 21 (1976), 43-63. Google Scholar
[5]	A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, No. 104, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260. Google Scholar
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[7]	J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin Heidelberg New York, 1976. Google Scholar
[8]	J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sc. Paris, 265 (1967), 333-336. Google Scholar
[9]	K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120. doi: 10.1016/0022-247X(80)90309-1. Google Scholar
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[11]	R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X. Google Scholar
[12]	K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005. Google Scholar
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[14]	J. Chazarain and A. Piriou, Introduction à La Théorie Des Équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981. Google Scholar
[15]	S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982. Google Scholar
[16]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar
[17]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. Google Scholar
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[37]	J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Differential Equations, 148 (1998), 47-64. doi: 10.1006/jdeq.1998.3456. Google Scholar
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[41]	T.-C. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^{p} = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124. Google Scholar
[42]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York London, 1992. Google Scholar
[43]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5. Google Scholar
[44]	P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202. doi: 10.1216/RMJ-1973-3-2-161. Google Scholar
[45]	R. Redlinger, Über die $C^{2}$-Kompaktheit der Bahn von Lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinburgh, 93 (1982/83), 99-103. doi: 10.1017/S0308210500031693. Google Scholar
[46]	M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York San Francisco London, 1978. Google Scholar
[47]	T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. Google Scholar
[48]	D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, No. 309, Springer-Verlag, New York Heidelberg Berlin, 1973. Google Scholar
[49]	J. C. Saut and B. Scheurer, Remarks on a non linear equation arising in population genetics, Comm. Partial Differential Equations, 3 (1978), 907-931. doi: 10.1080/03605307808820080. Google Scholar
[50]	M. Schechter, Principles of Functional Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, 36. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/gsm/036. Google Scholar
[51]	N. N. Semenov, Chemical Kinetics and Chain Reactions, Clarendon Press, Oxford, 1935. Google Scholar
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Figure 1. The bounded domain $ D $ and the unit outward normal $ \mathbf{n} $ to $ \partial D $

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Figure 2. The boundary portion $ M $ is deadly and its complement $ {\partial D} \setminus M $ is a barrier

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Figure 3. The structural condition (Z.1) on the function $ h(x) $

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Figure 4. The bifurcation diagram of Theorem 1.5

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Figure 5. The bifurcation diagram of Theorem 1.5: Malthus versus Verhulst

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Figure 6. The positive solution curve (16) for $ \lambda > \lambda_{1}(m) $ under condition (Z.3) via the Semenov approximation

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Figure 7. The bifurcation diagram of Remark 1.4 under condition (Z.3) (Verhulst theory)

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Figure 8. Conditions (b) and (d) in Theorem 3.3

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Figure 9. The bifurcation curves $ \varGamma_{1} $ and $ \varGamma_{2} $ of the nonlinear equation (20) in Theorem 3.3

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Figure 10. The point $ \left(1/ \mathop{\mathrm{spr}}(B), 0\right) $ is a bifurcation point of the nonlinear equation (21) to the trivial solution in Theorem 3.4

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Figure 11. The mapping properties of the resolvent $ R_{c} $ in the spaces $ C(\overline{D}) $, $ W^{2,p}(D) $ and $ C^{1}_{B}(\overline{D}) $

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Figure 12. The mapping properties of the resolvent $ R_{c} = \left(- \varDelta + c(x)\right)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $

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Figure 13. The first eigenvalues $ \mu_{1}(\lambda) = \gamma_{1}(\lambda) - \lambda $, $ \mu_{1}(0) = \gamma_{1} $ and $ \mu_{1}\left(\lambda_{1}(m)\right) = 0 $

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Figure 14. The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx < 0 $

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Figure 15. The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx = 0 $

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Figure 16. The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx > 0 $

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Figure 17. A flowchart of proof of Theorem 1.5, part (i)

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Figure 18. A flowchart of proof of Lemma 7.2

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Figure 19. The set of solutions of the semilinear problem (1) consists of a pitchfork near $ \lambda = \lambda_{1}(m) $

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Figure 20. The critical value $ \overline{\lambda}(h) $ of the positive bifurcation solution curve $ \mathcal{C} = \{(\lambda, u(\lambda))\} $

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Figure 21. The mapping properties of the resolvent $ R_{\lambda} = (\lambda I - \varDelta)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $

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Figure 22. A positive bifurcation solution curve $ (\lambda, u(\lambda)) $ of the nonlinear operator equation $ u = H(\lambda,u) $ can be continued beyond the point $ (\lambda^{\ast}, u^{\ast}) $ via the implicit function theorem (Theorem 3.1)

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Figure 23. The mapping properties of the negative Laplacian $ - \varDelta $ and the resolvent $ R_{0} = \left(- \varDelta\right)^{-1} $

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Figure 24. A flowchart of proof of Theorem 1.5, part (ii)

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Figure 25. The bifurcation diagram of Theorem 8.1 (the Dirichlet case)

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Figure 26. The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $ (the Neumann case)

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Figure 27. The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $ (the Neumann case)

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Figure 28. The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $ (the Neumann case)

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Figure 30. The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $

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Figure 31. The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $

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Figure 32. The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $

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Figure 29. The open subset $ D^{+} $ with boundary $ \partial D^{+} $

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Table 1. A biological meaning of each term

Term	Biological interpretation

D	Terrain
x	Location of the terrain
u(x)	Population density of a species inhabiting the terrain
∆	A member of the population moves about the terrain via the type of random walks occurring in Brownian motion
$\frac{1}{\lambda }$	Rate of diffusive dispersal
m(x)	Intrinsic growth rate
h(x)	Coefficient of intraspecific competition

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Term	Biological interpretation

D	Terrain
x	Location of the terrain
u(x)	Population density of a species inhabiting the terrain
∆	A member of the population moves about the terrain via the type of random walks occurring in Brownian motion
$\frac{1}{\lambda }$	Rate of diffusive dispersal
m(x)	Intrinsic growth rate
h(x)	Coefficient of intraspecific competition

Table 2. A biological meaning of boundary conditions

Boundary Condition	Biological interpretation

Dirichlet case (a(x') ≡ 0, b(x') ≡ 1)	Completely hostile (deadly) exterior
Neumann case (a(x') ≡ 1, b(x') ≡ 0)	Barrier
Robin or mixed-type case (a(x') + b(x') > 0)	Hostile but not completely deadly exterior

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Boundary Condition	Biological interpretation

Dirichlet case (a(x') ≡ 0, b(x') ≡ 1)	Completely hostile (deadly) exterior
Neumann case (a(x') ≡ 1, b(x') ≡ 0)	Barrier
Robin or mixed-type case (a(x') + b(x') > 0)	Hostile but not completely deadly exterior

Table 3. An overview of theorems for eigenvalue problems with indefinite weights

Problems	Conditions	Theorems

(6) (mixed type case)	(M.1) (H.1), (H.2)	Theorem 1.3 for λ₁(m)
(55) (Dirichlet case)	(M.1)	Theorem 6.1 for γ₁(m)
(58) (Neumann case)	(M.2)	Theorem 6.2 for ν₁(m)
(56), (59), (61)	(M.1), (M.2) (H.1), (H.2)	Theorem 6.3 for µ_D(λ), µ_N(λ), µ₁(λ)

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Problems	Conditions	Theorems

(6) (mixed type case)	(M.1) (H.1), (H.2)	Theorem 1.3 for λ₁(m)
(55) (Dirichlet case)	(M.1)	Theorem 6.1 for γ₁(m)
(58) (Neumann case)	(M.2)	Theorem 6.2 for ν₁(m)
(56), (59), (61)	(M.1), (M.2) (H.1), (H.2)	Theorem 6.3 for µ_D(λ), µ_N(λ), µ₁(λ)

Table 4. An overview of existence theorems for diffusive logistic problems

Problems	Conditions	Theorems

(1) (mixed type case)	(M.1) (Z.1), (Z.2) (H.1), (H.2)	Theorem 1.5 for u(λ)
(91) (Dirichlet case)	(M.1) (Z.1), (Z.2)	Theorem 8.1 for v(λ)
(93) (Neumann case)	(M.2) (Z.1), (Z.2)	Theorem 8.2 for w(λ)
(1), (91), (93)	(M.1), (M.2) (Z.1), (Z.2) (H.1), (H.2)	Theorem 8.3 for v(λ), w(λ), u(λ)
(1) (mixed type case)	(M.1), (Z.3) (H.1), (H.2)	Theorem 1.6 for u(λ)

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Problems	Conditions	Theorems

(1) (mixed type case)	(M.1) (Z.1), (Z.2) (H.1), (H.2)	Theorem 1.5 for u(λ)
(91) (Dirichlet case)	(M.1) (Z.1), (Z.2)	Theorem 8.1 for v(λ)
(93) (Neumann case)	(M.2) (Z.1), (Z.2)	Theorem 8.2 for w(λ)
(1), (91), (93)	(M.1), (M.2) (Z.1), (Z.2) (H.1), (H.2)	Theorem 8.3 for v(λ), w(λ), u(λ)
(1) (mixed type case)	(M.1), (Z.3) (H.1), (H.2)	Theorem 1.6 for u(λ)

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