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

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

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  • 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.

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


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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
     | Show Table
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    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)
    Robin or mixed-type case
    (a(x') + b(x') > 0)
    Hostile but not completely deadly exterior
     | Show Table
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    Table 3.  An overview of theorems for eigenvalue problems with indefinite weights

    Problems Conditions Theorems
    (mixed type case)
    (M.1) (H.1), (H.2) Theorem 1.3 for λ1(m)
    (Dirichlet case)
    (M.1) Theorem 6.1 for γ1(m)
    (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)
     | Show Table
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    Table 4.  An overview of existence theorems for diffusive logistic problems

    Problems Conditions Theorems
    (mixed type case)
    (Z.1), (Z.2)
    (H.1), (H.2)
    Theorem 1.5 for u(λ)
    (Dirichlet case)
    (Z.1), (Z.2)
    Theorem 8.1 for v(λ)
    (Neumann case)
    (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)
    (mixed type case)
    (M.1), (Z.3)
    (H.1), (H.2)
    Theorem 1.6 for u(λ)
     | Show Table
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