S-shaped Bifurcation Curve Research Articles

Effects of initial vegetation heterogeneity on competition of submersed and floating macrophytes

<p>Non-spatial models of competition between floating aquatic vegetation (FAV) and submersed aquatic vegetation (SAV) predict a stable state of pure SAV at low total available limiting nutrient level, <italic>N</italic>, a stable state of only FAV for high <italic>N</italic>, and alternative stable states for intermediate <italic>N</italic>, as described by an S-shaped bifurcation curve. Spatial models that include physical heterogeneity of the waterbody show that the sharp transitions between these states become smooth. We examined the effects of heterogeneous initial conditions of the vegetation types. We used a spatially explicit model to describe the competition between the vegetation types. In the model, the FAV, duckweed (<italic>L. gibba</italic>), competed with the SAV, Nuttall's waterweed (<italic>Elodea nuttallii</italic>). Differences in the initial establishment of the two macrophytes affected the possible stable equilibria. When initial biomasses of SAV and FAV differed but each had the same initial biomass in each spatial cell, the S-shaped bifurcation resulted, but the critical transitions on the <italic>N</italic>-axis are shifted, depending on FAV:SAV biomass ratio. When the initial biomasses of SAV and FAV were randomly heterogeneously distributed among cells, the vegetation pattern of the competing species self-organized spatially, such that many different stable states were possible in the intermediate <italic>N</italic> region. If <italic>N</italic> was gradually increased or decreased through time from a stable state, the abrupt transitions of non-spatial models were changed into smoother transitions through a series of stable states, which resembles the Busse balloon observed in other systems.</p>

An infinite semipositone problem with a reversed S-shaped bifurcation curve

<abstract><p>We study positive solutions to the two point boundary value problem:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{matrix}Lu = -u'' = \lambda \bigg\{\dfrac{A}{u^\gamma}+M\big[u^\alpha+u^\delta\big]\bigg\} \; ;\; (0, 1) \\ u(0) = 0 = u(1)\; \; \; \; \; \; \; \; \; \; \end{matrix} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ A < 0 $, $ \alpha \in (0, 1), \delta > 1, \gamma \in (0, 1) $ are constants and $ \lambda > 0, M > 0 $ are parameters. We prove that the bifurcation diagram $ (\lambda \text{ vs } \|u\|_\infty) $ for positive solutions is at least a reversed S-shaped curve when $ M\gg1 $. Recent results in the literature imply that for $ M\gg1 $ there exists a range of $ \lambda $ where there exist at least two positive solutions. Here, when $ M\gg1 $, we prove the existence of a range of $ \lambda $ for which there exist at least three positive solutions and that the bifurcation diagram is at least a reversed S-shaped curve. Further, via a quadrature method and Python computations, for $ M\gg1 $, we show that the bifurcation diagram is exactly a reversed S-shaped curve. Also, when the operator $ L $ is replaced by a $ p $-Laplacian operator with $ p > 1 $, as well as $ p $-$ q $ Laplacian operator with $ p = 4 $ and $ q = 2 $, we show that the bifurcation diagram is again an exactly reversed S-shaped curve when $ M\gg1 $.</p></abstract>

Existence of S-shaped type bifurcation curve with dual cusp catastrophe via variational methods

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form −Δpu=f(u), with p>1. We deal with relatively unexplored cases when f is non-Lipschitz at 0, f(0)=0 and f(u)<0, u∈(0,r), for some r<+∞. Using the nonlinear generalized Rayleigh quotients method we find a range of parameters where the equation may have distinct branches of solutions. As a consequence, applying the variational methods, we prove that the equation has at least three positive solutions with two of them linearly unstable and one linearly stable. The results evidence that the bifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophe. Our results are new even in the one-dimensional case and p=2.

Existence and multiplicity results for p-q-Laplacian boundary value problems

We study positive solutions to the boundary value problem $$ \displaylines{ -\Delta_p u - \Delta_q u = \lambda f(u) \quad \text{in } \Omega, \cr u = 0 \quad \text{on } \partial\Omega, }$$ where \(q \in (1,p)\) and &amp;Omega; is a bounded domain in \(\mathbb{R}^N\), \(N&gt;1\) with smooth boundary, \(\lambda\) is a positive parameter, and \(f:[0,\infty) \to (0,\infty)\) is \(C_1\), nondecreasing, and p-sublinear at infinity i.e. \(\lim_{t \to \infty} f(t)/t^{p-1}=0\). We discuss existence and multiplicity results for classes of such f. Further, when N=1, we discuss an example which exhibits S-shaped bifurcation curves.&#x0D; For more information see https://ejde.math.txstate.edu/special/01/a3/abstr.html

S-shaped bifurcation curve for a nonlocal boundary value problem

In this paper by using tools from bifurcation theory, particularly the Crandall–Rabinowitz bifurcation theorem, the Implicit Function Theorem and the persistence results of degenerate solutions, due to E.N. Dancer, a result will be proved for a S-shaped bifurcation curve of a nonlocal boundary value problem. The main results of this paper partially extend the main results in [11,12] to the nonlocal boundary value problems.

S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-III functional response

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response\begin{document}${\left\{ {\begin{array}{*{20}{l}} {{u^{\prime \prime }}(x) + \lambda \left[ {ru(1 - \frac{u}{q}) - \frac{{{u^p}}}{{1 + {u^p}}}\% } \right] = 0{\text{,}} - {\text{1}} &lt; x &lt; 1{\text{,}}} \\ {u( - 1) = u(1) = 0{\text{, }}} \end{array}} \right.},$\end{document}where u is the population density of the species, p &gt; 1, q, r are two positive dimensionless parameters, and λ &gt; 0 is a bifurcation parameter. For fixed p &gt; 1, assume that q, r satisfy one of the following conditions: (ⅰ) r ≤ η1, p* q and (q, r) lies above the curve\begin{document}$\begin{array}{l}{\Gamma _1} = \{ (q,r):q(a) = \frac{{a[2{a^p} - (p - 2)]}}{{{a^p} - (p - 1)}}{\rm{, }}\\\quad \quad \quad \quad \quad r(a) = \frac{{{a^{p - 1}}[2{a^p} - (p - 2)]}}{{{{({a^p} + 1)}^2}}}{\rm{, }}\sqrt[p]{{p - 1}}\% &lt; a &lt; C_p^*\} ;\end{array}$\end{document}(ⅱ) r ≤ η2, p* q and (q, r) lies on or below the curve Γ1, where η1, p* and η2, p* are two positive constants, and $C_{p}^{*}={{\left(\frac{{{p}^{2}}+3p-4+p\sqrt{%{{p}^{2}}+6p-7}}{4} \right)}^{1/p}}$. Then on the (λ, ||u||∞)-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of q, r and λ.

On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion

We study the bifurcation curve and exact multiplicity of positive solutions of a two-point boundary value problem arising in a theory of thermal explosion \begin{equation*} \left\{ \begin{array}{l} u^{\prime\prime}(x) + \lambda \exp ( \frac{au}{a+u}) =0, &nbsp; &nbsp -1 < x < 1, \\ u(-1)=u(1)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is the Frank--Kamenetskii parameter and $a>0$ is the activation energy parameter. By developing some new time-map techniques and applying Sturm's theorem, we prove that, if $a\geq a^{\ast \ast }\approx 4.107$, the bifurcation curve is S-shaped on the $(\lambda ,\Vert u \Vert _{\infty })$-plane. Our result improves one of the main results in Hung and Wang (J. Differential Equations 251 (2011) 223--237).

S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws

We study the bifurcation curve and exact multiplicity of positive solutionsof the combustion problem with general Arrhenius reaction-rate laws\begin{eqnarray}u^{\prime \prime }(x)+\lambda (1+\epsilon u)^{m}e^{\frac{u}{1+\epsilon u}}=0, -1 0$ and $-\infty < m <1$. We prove that, for $(-4.103\approx)$ $\tilde{m}\leq m < 1$ for someconstant $\tilde{m}$, the bifurcation curve is S-shaped on the $(\lambda, \|u\|_{\infty })$-plane if $0<\epsilon \leq \frac{6}{7}\epsilon _{\text{tr}}^{\text{Sem}}(m)$, where\begin{eqnarray}\epsilon _{\text{tr}}^{\text{Sem}}(m)=\left\{\begin{array}{l}(\frac{1-\sqrt{1-m}}{m})^{2}\ \text{ for }-\infty < m < 1, m \neq 0, \\\frac{1}{4}\ \text{for}\ m=0,\end{array}\right.\end{eqnarray}is the Semenov transitional value for general Arrhenius kinetics. Inaddition, for $-\infty < m < 1$, the bifurcation curve is S-like shaped if $0<\epsilon \leq \frac{8}{9} \epsilon _{\text{tr}}^{\text{Sem}}(m).$ Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031--1048.)

S-shaped bifurcation curves for nonlinear two-parameter problems

We consider the nonlinear eigenvalue problem −u″(t)=λ(1+u(t)+u(t)2−ϵu(t)p),u(t)>0,t∈I:=(−1/2,1/2),u(−1/2)=u(1/2)=0, where p>2 is a constant, λ>0 and ϵ>0 are parameters. This equation was introduced by Crandall and Rabinowitz (1973) as the typical model which develops the S-shaped bifurcation curve when p=3 and 0<ϵ≪1. It is known that if p>2 and 0<ϵ≪1, then there exists a unique solution (λϵ(α),uϵ,α)∈R+×C2(Ī) with ‖uϵ,α‖∞=α, where α>0 is a given constant. Furthermore, λ=λϵ(α) is S-shaped. In this paper, we establish the asymptotic formulas for λ=λϵ(α) as α→0,α1,ϵ,α2,ϵ,βϵ when 0<ϵ≪1, where α1,ϵ and α2,ϵ are two turning points of λϵ(α), and βϵ is the unique positive solution to the algebraic equation 1+x+x2−ϵxp=0. We also characterize the asymptotic shape of uϵ,α(t) as α→βϵ by establishing the asymptotic behavior of ‖uϵ,α‖q as α→βϵ when 0<ϵ≪1 is a fixed constant.

Weak Allee effect, grazing, and S-shaped bifurcation curves

We study a one-dimensional reaction-diffusion model arising in population dynamics where the growth rate is a weak Allee type. In particular, we consider the effects of grazing on the steady states and discuss the complete evolution of the bifurcation curve of positive solutions as the grazing parameter varies. We obtain our results via the quadrature method and Mathematica computations. We establish that the bifurcation curve is S-shaped for certain ranges of the grazing parameter. We also prove this occurrence of an S-shaped bifurcation curve analytically.

Existence of Alternate Steady States in a Phosphorous Cycling Model

We analyze the positive solutions to the steady-state reaction diffusion equation with Dirichlet boundary conditions of the form: . Here, is the Laplacian of , is the diffusion coefficient, and are positive constants, and is a smooth bounded region with in . This model describes the steady states of phosphorus cycling in stratified lakes. Also, it describes the colonization of barren soils in drylands by vegetation. In this paper, we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of subsuper solutions.

Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem

We study the bifurcation curve of positive solutions of the combustion problem with nonlinear boundary conditions given by{−u″(x)=λexp(βuβ+u),0<x<1,u(0)=0,u(1)u(1)+1u′(1)+[1−u(1)u(1)+1]u(1)=0, where λ>0 is called the Frank–Kamenetskii parameter or ignition parameter, β>0 is the activation energy parameter, u(x) is the dimensionless temperature, and the reaction term exp(βuβ+u) is the temperature dependence obeying the simple Arrhenius reaction-rate law. We prove rigorously that, for β>β1≈6.459 for some constant β1, the bifurcation curve is double S-shaped on the (λ,‖u‖∞)-plane and the problem has at least six positive solutions for a certain range of positive λ. We give rigorous proofs of some computational results of Goddard II, Shivaji and Lee [J. Goddard II, R. Shivaji, E.K. Lee, A double S-shaped bifurcation curve for a reaction–diffusion model with nonlinear boundary conditions, Bound. Value Probl. (2010), Art. ID 357542, 23 pp.].

A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem

We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem { u ″ ( x ) + λ f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter, f ∈ C 2 [ 0 , ∞ ) satisfies f ( 0 ) > 0 and f ( u ) > 0 for u > 0 , and f is convex–concave on ( 0 , ∞ ) . Under a mild condition, we prove that the bifurcation curve is S-shaped on the ( λ , ‖ u ‖ ∞ ) -plane. We give an application to the perturbed Gelfand problem { u ″ ( x ) + λ exp ( a u a + u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where a > 0 is the activation energy parameter. We prove that, if a ⩾ a ⁎ ≈ 4.166 , the bifurcation curve is S-shaped on the ( λ , ‖ u ‖ ∞ ) -plane. Our results improve those in [S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475–1485] and [P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999) 1011–1020].

S-shaped bifurcation curves in ecosystems

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: { − Δ u = λ [ u − u 2 K − c u 2 1 + u 2 ] , x ∈ Ω , u = 0 , x ∈ ∂ Ω . Here Δ u = div ( ∇ u ) is the Laplacian of u, 1 λ is the diffusion coefficient, K and c are positive constants and Ω ⊂ R N is a smooth bounded region with ∂ Ω in C 2 . This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions.

On S-shaped and reversed S-shaped bifurcation curves for singular problems

We analyze the positive solutions to the singular boundary value problem ( −(|u ′ | p−2 u ′ ) ′ = � g(u) u� ; (0, 1), u(0) = 0 = u(1), where p > 1,� ∈ (0, 1),� > 0 and g : [0, ∞) → R is a C 1 function. In particular, we discuss examples when g(0) > 0 and wheng(0) < 0 that lead to S-shaped and reversed S-shaped bifurcation curves, respectively.

A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions

We study the positive solutions to boundary value problems of the form ; , ; , where is a bounded domain in with , is the Laplace operator, is a positive parameter, is a continuous function which is sublinear at , is the outward normal derivative, and is a smooth function nondecreasing in . In particular, we discuss the existence of at least two positive radial solutions for when is an annulus in . Further, we discuss the existence of a double S-shaped bifurcation curve when , , and with .

On S-shaped bifurcation curves for a class of perturbed semilinear equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.

A theorem on reversed S-shaped bifurcation curves for a class of boundary value problems and its application

We study the bifurcation curves of positive solutions of the boundary value problem { u ″ ( x ) + f ε ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where f ε ( u ) = g ( u ) − ε h ( u ) , ε ∈ R is a bifurcation parameter, and functions g , h ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) satisfy five hypotheses presented herein. Assuming these hypotheses on fixed g and h , we prove that the bifurcation curve is reverse S-shaped on the ( ε , ‖ u ‖ ∞ ) -plane; that is, the bifurcation curve has exactly two turning points at some points ( ε ̃ , ‖ u ε ̃ ‖ ∞ ) and ( ε ∗ , ‖ u ε ∗ ‖ ∞ ) such that ε ̃ < ε ∗ and ‖ u ε ̃ ‖ ∞ < ‖ u ε ∗ ‖ ∞ . In addition, we prove that ε ∗ > 0 . Thus the exact number of positive solutions can be precisely determined by the values of ε ̃ and ε ∗ . We give an application to the two-parameter bifurcation problem { u ″ ( x ) + λ ( 1 + u 2 − ε u 3 ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ , ε are two positive bifurcation parameters. Some new results are obtained.

Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors

Abstract In this paper, we perform linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors. We consider three different models of varying dimensionality—the 3D convection-diffusion-reaction (CDR) model is the high dimensional one, and the Liapunov–Schmidt reduction based spatially averaged two-dimensional CDR model and its regularized form are the two low-dimensional ones. For each of these three models, steady state bifurcation diagrams that show the presence of multiple steady states were obtained and the stability of these multiple steady states to transverse perturbations was analyzed using linear stability analysis. Parametric analysis of the steady state bifurcation diagrams shows that for sufficiently large values of transverse Peclet number p, mixing-limited patterns may emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Comparison of the bifurcation diagrams and the stability boundaries of the two low-dimensional models with that of the 3D CDR model reveals that the regularized form of the low-dimensional model has higher accuracy and a larger region of validity than the averaged form and is therefore recommended over the latter.

Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions

Interaction between transport and reaction generates a variety of complex spatio-temporal patterns in chemical reactors. These patterned states, which are typically initiated by autocatalytic effects and sustained by differences in diffusion/local mixing rates, often cause undesired effects in the reactor. In this work, we analyze the dynamic evolution of mixing-limited spatial pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors using two-dimensional (2-D) convection-diffusion-reaction (CDR) models that are obtained through rigorous spatial averaging of the three-dimensional (3-D) CDR model using Liapunov-Schmidt technique of bifurcation theory. We use the spatially-averaged 2-D CDR model (and its "regularized" form) to perform steady-state bifurcation analysis that captures the region of multiple solutions, and we analyze the stability of these multiple steady states to transverse perturbations using linear stability analysis. Parametric analyses of the steady-state bifurcation diagrams and stability boundaries show that when transverse mixing is significantly slower than the rate of autocatalytic reaction, mixing-limited patterns emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Our dynamic simulations show the emergence of three different types of spatial patterns namely, Band, Anti-phase and Target, depending on the nature of transverse perturbation. The temporal evolution of these patterns consists of rapid intensification of the concentration-segregation process (especially when transverse mixing is much slower than reaction) followed by slow diffusion-mediated return to symmetry that occurs at time scales much larger than the reactor residence time. Our parametric analysis of the dynamics reveals that while larger Péclet numbers (both axial and transverse) increase the stability and decay time of the patterned states, larger Damköhler numbers lead to faster ignition resulting in the opposite effect.