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doi: 10.3934/dcdss.2021166
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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.

Citation: Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021166
References:
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H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, North-Holland Math. Studies, 21 (1976), 43-63. 

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show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, Vol. 140, Elsevier/Academic Press, Amsterdam, 2003.

[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.

[3]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[4]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, North-Holland Math. Studies, 21 (1976), 43-63. 

[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.
[6] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, No. 34, Cambridge University Press, Cambridge, 1995. 
[7]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin Heidelberg New York, 1976.

[8]

J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sc. Paris, 265 (1967), 333-336. 

[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.

[10]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.

[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.

[12]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.

[13] I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. 
[14]

J. Chazarain and A. Piriou, Introduction à La Théorie Des Équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.

[15]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.

[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.

[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.

[18]

E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192.  doi: 10.1007/BF00282326.

[19]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential Equations, 957 (1982), 34-87. 

[20]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[21]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, 2$^{nd}$ edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0387-8.

[22]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151.

[23]

J. M. FraileP. Koch MedinaJ. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.  doi: 10.1006/jdeq.1996.0071.

[24]

J. García-MeliánR. Gómez-ReñascoJ. López-Gómez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rational Mech. Anal., 145 (1998), 261-289.  doi: 10.1007/s002050050130.

[25]

J. García-MeliánJ. D. Rossi and J. C. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Commun. Contemp. Math., 11 (2009), 585-613.  doi: 10.1142/S0219199709003508.

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[27]

I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk., 12 (1957), 43–118; English translation: Amer. Math. Soc. Transl., 13 (1960), 185–264. doi: 10.1090/trans2/013/08.

[28]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematical Series, 247, Longman Scientific & Technical, Harlow, New York, 1991.

[29]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.  doi: 10.1080/03605308008820162.

[30]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274. Springer-Verlag, Berlin, 1994.

[31]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964.

[32]

M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 1950 (1950), 128 pp.

[33]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, Rhode Island 1968.

[34] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1964 (Russian), English translation; Academic Press, New York London, 1968. 
[35]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics, A dream of Volterra, Elsevier/North-Holland, Amsterdam, Stationary Partial Differential Equations, 2 (2005), 211-309.  doi: 10.1016/S1874-5733(05)80012-9.

[36]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.

[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.

[38]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellitici del secondo ordine, Boll. Un. Mat. Ital., 7 (1973), 285-301. 

[39]

J. Moser, A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.

[40]

L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University, Courant Institute of Mathematical Sciences, New York; revised reprint of the 1974 original, Courant Lecture Notes in Mathematics, No, 6, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/cln/006.

[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.

[42] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York London, 1992. 
[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.

[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.

[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.

[46] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York San Francisco London, 1978. 
[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.

[48]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, No. 309, Springer-Verlag, New York Heidelberg Berlin, 1973.

[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.

[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.

[51] N. N. Semenov, Chemical Kinetics and Chain Reactions, Clarendon Press, Oxford, 1935. 
[52]

S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics, Comm. Partial Differential Equations, 8 (1983), 1199-1228.  doi: 10.1080/03605308308820300.

[53]

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Figure 1.  The bounded domain $ D $ and the unit outward normal $ \mathbf{n} $ to $ \partial D $
Figure 2.  The boundary portion $ M $ is deadly and its complement $ {\partial D} \setminus M $ is a barrier
Figure 3.  The structural condition (Z.1) on the function $ h(x) $
Figure 4.  The bifurcation diagram of Theorem 1.5
Figure 5.  The bifurcation diagram of Theorem 1.5: Malthus versus Verhulst
Figure 6.  The positive solution curve (16) for $ \lambda > \lambda_{1}(m) $ under condition (Z.3) via the Semenov approximation
Figure 7.  The bifurcation diagram of Remark 1.4 under condition (Z.3) (Verhulst theory)
Figure 8.  Conditions (b) and (d) in Theorem 3.3
Figure 9.  The bifurcation curves $ \varGamma_{1} $ and $ \varGamma_{2} $ of the nonlinear equation (20) in Theorem 3.3
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
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}) $
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}) $
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 $
Figure 14.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx < 0 $
Figure 15.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx = 0 $
Figure 16.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx > 0 $
Figure 17.  A flowchart of proof of Theorem 1.5, part (i)
Figure 18.  A flowchart of proof of Lemma 7.2
Figure 19.  The set of solutions of the semilinear problem (1) consists of a pitchfork near $ \lambda = \lambda_{1}(m) $
Figure 20.  The critical value $ \overline{\lambda}(h) $ of the positive bifurcation solution curve $ \mathcal{C} = \{(\lambda, u(\lambda))\} $
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}) $
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)
Figure 23.  The mapping properties of the negative Laplacian $ - \varDelta $ and the resolvent $ R_{0} = \left(- \varDelta\right)^{-1} $
Figure 24.  A flowchart of proof of Theorem 1.5, part (ii)
Figure 25.  The bifurcation diagram of Theorem 8.1 (the Dirichlet case)
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)
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)
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)
Figure 30.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $
Figure 31.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $
Figure 32.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $
Figure 29.  The open subset $ D^{+} $ with boundary $ \partial D^{+} $
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
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
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 µD(λ), µN(λ), µ1(λ)
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 µD(λ), µN(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)
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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