Partial differential equation theory, especially the theory of reaction-diffusion equations, has long been found extremely useful in the qualitative study of theoretical ecology. The interplay between partial differential equations and mathematical ecology not only benefits ecology greatly, it is also one of the important sources from which many beautiful and challenging mathematical models have been derived and henceforth helps to develop the theory of partial differential equations. In this thesis, we are concerned with the mathematical study of a basic model proposed by ecologists describing the ecological behaviors of phytoplankton in eutrophic environments. The model is viewed by some ecologists as a very realistic model to describe the behavior of phytoplankton. On the other hand, the model involves a nonlocal term and is mathematically challenging, and not much mathematical research has been carried out on it. This thesis is an attempt to further the understanding of this model

Du, Yihong

Harris, Adam

Mei, Linfeng

Yan, Shusen

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REACTION DIFFUSION EQUATIONSARISING FROM PHYTOPLANKTONDYNAMICSByLinfeng MeiMaster of Science, Shandong UniversitySchool of Science and TechnologyUniversity of New EnglandA thesis submitted for the degree ofDoctor of Philosophyof University of New EnglandFebruary 2012AcknowledgementsFirst of all, I would like to thank my principal supervisor, Professor Yihong Du, for his patientguidance, ever continuing support over the past three years, without which I could not havecompleted this dissertation. I am also deeply grateful to the co-supervisor, Associate ProfessorShusen Yan, for all his support and warm encouragement, which greatly boosted my confidence.Special thanks also go to Dr Adam Harris, who is the co-supervisor while Professor Shusen Yanis away, for his kindness and support.I would also like to thank the staff in our department, Dr. Bea Bleile, Dr. Gerd Schmalz, Dr.Imre Boker and others for their kind support. I cherish every moment of my wonderful stay inArmidale, which will definitely become an unforgettable memory in my life.The research of this dissertation has been undertaken with the support of a university post-graduate research scholarship of the University of New England and an international postgrad-uate research scholarship of the Australian government. This dissertation would not have beencompleted without these fundings.iiiPublicationsJournal papers published1. Wei Dong, Linfeng Mei, Multiple solutions for an indefinite superlinear elliptic problemon RN , Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 2056-2070.2. Yihong Du, Linfeng Mei, On a nonlocal reaction-diffusion-advection equation modellingphytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.3. Linfeng Mei, Xiaoyan Zhang, On a nonlocal reaction-diffusion-advection system model-ing phytoplankton growth with light and nutrients, Discrete and Continuous DynamicalSystems - Series B, 17 (2012), 221-243.4. Linfeng Mei, Xiaoyan Zhang, Existence of positive stationary solutions for a diffusivevariable-territory predator-prey model, IMA Journal of Applied Mathematics, publishedonline December 2011, doi:10.1093/imamat/hxr059.Papers submitted/in preparation1. Linfeng Mei, Xiaoyan Zhang, Coexistence and competitive exclusion in multi-species phy-toplankton dynamics, submittedivAcademic activities(1) The Inaugural Pacific Rim Mathematical Association (PRIMA) Congress, held in Univer-sity of New South Wales, Sydney, Australia, July 6-10, 2009.(2) 20 minutes contributed talk in Workshop On PDE Models Of Biological Process, held inNational Hsing-Hua University, Taiwan, December 13-17, 2010.Talk title: Limiting profiles of a phytoplankton model.(3) The 8th East China Partial Differential Equations Conference & The 2th InternationalWorkshop On Reaction-Diffusion Models And Mathematical Biology, held in Xi’an, Chi-na, July 11-14.(4) 30 minutes contributed talk in the 55th Annual Meeting of the Australian MathematicalSociety, University of Wollongong, Wollongong, Australia, September 26-29, 2011.Talk Title: On a nonlocal equation modeling phytoplankton growth.vAbstractPartial differential equation theory, especially the theory of reaction-diffusion equations, has longbeen found extremely useful in the qualitative study of theoretical ecology. The interplay be-tween partial differential equations and mathematical ecology not only benefits ecology greatly,it is also one of the important sources from which many beautiful and challenging mathemati-cal models have been derived and henceforth helps to develop the theory of partial differentialequations.In this thesis, we are concerned with the mathematical study of a basic model proposedby ecologists describing the ecological behaviors of phytoplankton in eutrophic environments.The model is viewed by some ecologists as a very realistic model to describe the behavior ofphytoplankton. On the other hand, the model involves a nonlocal term and is mathematicallychallenging, and not much mathematical research has been carried out on it. This thesis is anattempt to further the understanding of this model. It consists of five chapters.In Chapter 1, we collect some basic mathematical facts that the thesis will be based on.These facts include the existence and uniqueness theorem of ordinary differential equations, thedefinitions of Hölder spaces and Sobolev spaces and the embedding theorems, the Lp estimatesof partial differential equations, the principal eigenvalues and the maximum principles, as wellas topological degree theory and bifurcation theorems.In Chapter 2, we first give a brief introduction on the biological background of our modeland its mathematical formulation. Then we recall the existing mathematical works on the modeland briefly outline the work we have done.viIn Chapter 3. We study the one species model:ut = Jx(x, t) +[g(e−k0x−k∫ x0 u(s,t)ds)− d]u, 0 < x < h, t > 0,J(x, t) = D(x)ux(x, t)− α(x)u(x, t) = 0, x = 0 or h, t > 0,u(x, 0) = u0(x) 	 0, 0 ≤ x ≤ h,where g(I) is a smooth function satisfyingg(0) = 0, g′(I) > 0 for I > 0,and for any γ > 0 ∫ ∞0g(e−γx)dx <∞.We prove there is a critical d∗ such that when 0 < d < d∗, the above equation has a uniquepositive steady state u∗d, and when d ≥ d∗, it has no positive steady state. Moreover, when0 < d < d∗, u∗d is the global attractor, when d ≥ d∗, 0 is the global attractor. We also prove inthis chapter that when the diffusion coefficient approaches zero the phytoplankton species willbehave like a δ-function, concentrating at the bottom of the water column; when the diffusioncoefficient is very large, the phytoplankton species will distribute evenly in the water column. Weaslo study in this chapter the behavior of the phytoplankton species when the depth of the watercolumn goes to infinite, and find that the distribution of the phytoplankton species converges tothat studied by Ishii and Takagi [40]. As a byproduct, we solved a problem left open in Hsu andLou [33].In Chapter 4 and Chapter 5, we study the behavior of multiple species phytoplankton that livein the same vertical water column:(ui)t = (Di(x) (ui)x − αi(x)ui)x + (gi(I(x, t))− di)ui, i = 1, · · · , n, (0.0.1)with zero-flux boundary conditionsDi(x)(ui)x(x, t)− αi(x)ui(x, t) = 0, x = 0, h, t ≥ 0, i = 1, · · · , n,and initial conditionsui(x, 0) = u0i (x) 	 0, 0 ≤ x ≤ h, i = 1, · · · , n,viigi(I) satisfiesgi(0) = 0, g′i(I) > 0 for I ≥ 0,and there are positive constants ci, γi such thatgi(I) ≤ ciIγi for any I ≥ 0,I(x, t) = I0e−k0x exp(−∫ x0[k1u1(s, t) + · · ·+ knun(s, t)] ds).In Chapter 4, we study the existence of positive steady states for the two species model.Among other things, we use the topological degree argument to find a group of sufficient condi-tions under which the model has at least one positive steady state. We also study the necessaryconditions for the existence of positive steady states and show that the sufficient conditions arenot necessary.In Chapter 5, we study the existence of positive steady state of the more than two speciesmodel. As in Chapter 4, we find a group of sufficient conditions for the existence of the positivesteady state for the multiple species mode in terms of principal eigenvalues:0 < di < −λ(i)1 [−gi(κi(x))], i = 1, 2, · · · , n, (0.0.2)whereκi(x) = e−k0x exp(∑j 6=i∫ x0udj(y)dy), i = 1, 2, · · · , n,λ(i)1 (Ψ) is the principal eigenvalue of the eigenvalue problem−Djφ′′ + αiφ′ + Ψφ = λφ, Djφ′(0)− αjφ′(0) = Djφ′(1)− αjuφ′(1) = 0,and udj is the unique positive solution of the equation−Dju′′ + αju′ =[gj(e−k0x−∫ x0 u(y)dy)− dj]u, Dju′(0)− αju′(0) = Dju′(1)− αju′(1) = 0.Here we assume for simplicity, Dj(x) ≡ Dj, αj(x) ≡ αj for j = 1, · · · , n. The derivation ofthis sufficient condition is more complicated, and cannot use the topological degree argumentin Chapter 4 directly. It is derived from a fixed point index calculation technique developed byDancer and Du in [14].viiiA considerable part of Chapter 5 is devoted to the explanation of the rather implicit sufficientcondition (0.0.2). By developing an idea from Hsu and Lou [33], we use the fine propertiesof the eigenvalues to find two groups of explicit conditions under which (0.0.2) is satisfied.Interestingly, one group of conditions require the diffusions of the system to be sufficiently small,as opposed to the situations that when the diffusions of the system is sufficiently large, the systemcannot have any positive steady states. This explains from one angle the famous paradox ofplankton.ixContentsAcknowledgements iiiPublications ivAcademic activities vAbstract vi1 Preliminaries 11.1 Basic theory of differential equations . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The existence and uniqueness theorem of ODE . . . . . . . . . . . . . . 11.1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Basic theory of second order linear elliptic and parabolic equations . . . 41.2 Principal eigenvalues and maximum principles . . . . . . . . . . . . . . . . . . 61.3 Topological degree and bifurcation theorems . . . . . . . . . . . . . . . . . . . . 92 Background and outline of research 122.1 The models and existing mathematical works . . . . . . . . . . . . . . . . . . . 122.2 Outline of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 The one species case 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Existence and uniqueness of positive steady state solution . . . . . . . . . . . . . 263.3 The global stability of the corresponding parabolic equation . . . . . . . . . . . 35x3.4 The limiting profile of the positive steady state solution . . . . . . . . . . . . . . 453.4.1 The small diffusion case . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.2 The large diffusion case . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.3 The deep water column case . . . . . . . . . . . . . . . . . . . . . . . . 554 The two species case 704.1 Steady-states of the two species model . . . . . . . . . . . . . . . . . . . . . . . 705 The more than two species case 825.1 The multi-species case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Existence and non-existence of positive steady state solutions . . . . . . . . . . . 855.3 The proof of Theorem 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 The n-species case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xi

Reaction Diffusion Equations Arising from Phytoplankton Dynamics

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Reaction Diffusion Equations Arising from Phytoplankton Dynamics

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