Abstract:

From logistic equation to Lotka-Volterra competition system, the interaction between spatial heterogeneity and diffusion creates unexpected phenomena. This has led ecologists as well as biologists to conduct experiments to verify if the counter intuitive mathematical results could actually take place in ecology. An affirmative answer has been obtained recently and this has opened up a new direction in ecology with many possible consequences. One of the important by-products for mathematics is that, based on the experiments, revisions have been proposed to bring the original mathematical models and equations closer to reality which, inevitably, poses serious mathematical challenges. In this course, I will describe some of the recent developments in this direction.

Abstract:

Some invasive species can persist in their habitat but eventually spread very slow in a nonlinear fashion to expand their habitat range. To capture this phenomenon, we consider reaction-diffusion-advection models with a free boundary to model the spreading of species in one-dimensional spatially heterogeneous environments. We shall investigate the effect of the biased movement strategy and the resource on the dynamics.

Abstract:

An initial boundary value problem for a reaction-diffusion system arising in the study of a singular predator-prey system shall be addressed. First, we shall talk about the associated kinetic system. In particular, we shall concentrate on the case when the unique co-existence state is a center for the kinetic system. Then we shall investigate solutions of the diffusion system with equal diffusivity. An interesting phenomenon, spatio-temporal oscillation (namely, solutions become spatially homogeneous and are subject to the kinetic part asymptotically) for the reaction-diffusion system, shall be described and derived.

Abstract:

Propagation phenomena are often observed in many fields including dissipative situations. To characterize the universal profiles of these phenomena, traveling wave solutions and entire solutions play important roles. Here traveling wave solution is meant by a solution of a partial differential equation that propagates with a constant speed, while it maintains its shape in space, and an entire solution is a solution defined for all time.

In this series of talks, we mainly focus on the Allen-Cahn-Nagumo equation, which is a single reaction diffusion equation with bi-stable nonlinearity. One of the ways to handle more complicated propagation phenomena is to derive a method to compose the known solutions. I will explain how to construct entire solutions by composing the traveling wave solutions. I will also discuss the relation between traveling wave solutions and entire solutions. This observation also suggests us the existence of new types of entire solutions.

Abstract:

Activator-inhibitor models are used to explain pattern formation and wave propagation in nature. One of the prominent representatives of these models is the FitzHugh-Nagumo system which describes the mathematical properties of electrochemical interactions in neurons and cardiac tissues. This series of lectures focus on the standing and traveling fronts/pulses of the FitzHugh-Nagumo system and related activator-inhibitor models with skew-gradient structure. We will introduce a variational formulation involving a nonlocal term to study these wave solutions. For the standing solutions, the energy functional in such a formulation is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulses. To overcome this difficulty, one needs to consider a local minimizer extracted from a suitable topological class of admissible functions. For the study of a traveling front/pulse, an additional parameter is introduced in the non-local term of the variational energy. We will explain how to determine the speed of a traveling front/pulse via the choice of the parameter and to prove the existence of such front/pulse by minimizing the energy subject to a constraint.

Abstract:

In the fields of population biology and cell biology concentration phenomena are often observed by aggregation of species and chemical substances respectively. One of the well-known models is a Keller-Segel chemotaxis model [21]in which spiky patterns appears by the aggregation of cellular slime mold, though it blows up in a higher dimensional domain (for instance, see [17], [5], [23], [20], [25]and the references therein). In this model the total mass of the slime mold is conserved in a reasonable setting. On the other hand in a study for the cell polarity the authors [19] and [7] proposed simple conceptual models to describe the concentration phenomenon induced by a different mechanism from the chemotaxis model, though the mass conservation property shares in the both models. After their contribution, mathematical studies for the conceptual models are developed in [16], [15], [8], [10]and [9] (see also [13], [14], [11]and [12]). In particular, it is shown in [16], [15]and [8] that the spiky pattern is certainly stable in their model equations.

Motivated by those studies, we are concerned with stationary problem of a reaction–diffusion system with a conservation law under the Neumann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coefficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling withκthe asymptotic profile exhibits a parabola in the non-vanishing region. We will also study the existence of an unstable monotone solution when the mass is small.

By the way, in this course, we will also discuss some interesting open problems in high dimensions.

References:

[1] S. Boussaïd, D. Hilhorst, T.N. Nguyen, Convergence to steady state for the solutions of a nonlocal reaction–diffusion equation, Evol. Equ. and Control Theory 4 (2015) 39–59.

[2] J. Carr, M.E. Gurtin, M. Slemrod, Structured phase transitions on a finite interval, Arch. Ration. Mech. Anal. 86 (1984) 317–351.

[3] J.-L. Chern, Y. Morita, T.-T. Shieh, Asymptotic behavior of equilibrium states of reaction-diffusion systems with mass conservation.J. Differential Equations264 (2018), no. 2,550–574.

[4] M.E. Gurtin, H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 156 (1988) 301–317.

[5] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol.25, American Mathematical Society, Providence, RI, 1988.

[6] M. Herrero, J. Velzquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583–623.

[7] S.-Z. Huang, Gradient Inequalities: With Applications to Asymptotic Behavior and Stability of Gradient-Like Systems, Math. Surveys Monogr., vol.126, American Mathematical Society, Providence, RI, 2006.

[8] S. Ishihara, M. Otsuji, A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E 75 (2007), 015203(R).

[9] S. Jimbo, Y. Morita, Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation, J. Differential Equations 255 (2013) 1657–1683.

[10] E. Latos, Y. Morita, T. Suzuki, Stability and spectral comparison of a reaction-diffusion system with mass conservation, preprint.

[11] E. Latos, T. Suzuki, Global dynamics of a reaction–diffusion system with mass conservation, J. Math. Anal. Appl. 411 (2014) 107–118.

[12] T. Mori, K. Kuto, T. Tsujikawa, M. Nagayama, S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction–diffusion model for cell polarization, in: Dynamical Systems, Differential Equations and Applications AIMS Proceedings, 2015, pp.861–877.

[13] T. Mori, K. Kuto, T. Tsujikawa, S. Yotsutani, Exact multiplicity of stationary limiting problem of a cell polarization model, Discrete Contin. Dyn. Syst. Ser. A 36 (2016) 5627–5655.

[14] Y. Mori, Y. Jilkine, L. Edelstein-Keshet, Wave-pinning and cell polarity from bistable reaction–diffusion system, Biophys. J. 94 (2008) 3684–3697.

[15] Y. Mori, Y. Jilkine, L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction–diffusion model for cell polarization, SIAM J. Appl. Math. 71 (2011) 1401–1427.

[16] Y. Morita, Spectrum comparison for a conserved reaction–diffusion system with a variational property, J. Comput. Anal. Appl. 2 (2012) 57–71.

[17] Y. Morita, T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction–diffusion system with conservation of mass, Nonlinearity 23 (2010) 1387–1411.

[18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581–601.

[19] A. Novick-Cohen, On the viscous Chan–Hilliard equation, in: J.M. Ball (Ed.), Material Instabilities in Continuum Mechanics and Related Mathematical Problems, Clarendon Press, Oxford, 1988, pp.329–342.

[20] M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki, S. Kuroda, A mass conserved reaction–diffusion system captures properties of cell polarity, PLoS Comput. Biol. 3 (2007) 1040–1054.

[21] T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217.

[22] E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415.

[23] K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H.M. Byrne, V. Cristini, J. Lowengrub, Density-dependent quiescence in glioma invasion: instability in a simple reaction–diffusion model for the migration/proliferation dichotomy, J.Biol. Dyn. 6 (2012) 54–71.

[24] T. Senba, T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal. 8 (2001) 349–367.

[25] T. Suzuki, S. Tasaki, Stationary Fix–Caginalp equation with non-local term, Nonlinear Anal. 71 (2009) 1329–1349.

[26] X. Wang, Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, J. Math. Biol. 66 (2013) 1241–1266.

Abstract:

Understanding ion transport is crucial in the study of many physical and biological problems, such as semiconductors, electro-kinetic fluids, and ion channels in cell membranes. One of the fundamental models for the ionic transport is the time dependent coupled diffusion-convection equations, the Poisson-Nernst-Planck (PNP) system. However, due to the lack of ionic finite size effects, conventional PNP systems cannot be used to study gating and selectivity of ion channels and new mathematical models need to be developed. In my lectures, I’ll introduce analytical techniques to derive and analyze PNP type systems for the study of ion transport through channels.

Abstract:

In this 3-hour mini-course, we shall introduce the abstract monotone dynamical systems with the examples in Ecology, in particular the cooperative ODE systems and reaction-diffusion systems and two species competitive systems in ordered Banach Space.

We also introduce the Poincare-Bendixson Theorem for 3-dimension K-type competitive systems with examples in two species competitive systems with drug resistance. Several open problems are proposed.

Abstract:

Competition for resources is a fundamental interaction between species and there has been a lot of experimental and theoretical analyses of nutrient-limited phytoplankton growth and competition studies. The simplest competition models neglect differences between individuals, using one ordinary differential equation to govern the dynamics of each species. These population dynamics are coupled to dynamics of one or more resources by assuming a constant quota of nutrient per individual, or equivalently, a constant yield of individuals from consumption of a unit of resource. In fact, quotas may vary, leading to variable-internal-stores models.
Ecologists believe that spatial or temporal variations can alter competitive outcomes, perhaps promoting coexistence of species and diversity in competitor communities.
In this talk, I shall discuss some systems modeling resource(s) consumption, storage, and population growth in spatially varying environments (PDEs system) or temporally varying environments (ODEs system involved in time-periodic coefficients).

References:

[1] Sze-Bi Hsu, Jifa Jiang and Feng-Bin Wang, On a system of reaction–diffusion equations arising from competition with internal storage in an unstirred chemostat, Journal of Differential Equations 248 (2010), pp. 2470–2496.
[2] James P. Grover, Sze-Bi Hsu and Feng-Bin Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, Journal of Mathematical Biology, Vol. 64 (2012), pp. 713–743.
[3] Sze-Bi Hsu, Feng-Bin Wang and Xiao-Qiang Zhao, Competition for two essential resources with internal storage and periodic input, Differential and Integral Equations, Vol. 29, 2016, pp. 601-630.
[4] Sze-Bi Hsu,King-Yeung Lam, and Feng-Bin Wang, Single species growth  consuming inorganic carbon with internal storage in a poorly mixed habitat, Journal of Mathematical Biology, Vol. 75, Dec., 2017, pp.1775-1825.