Reaction-Diffusion and Mathematical Sciences
Every Sunday 9:45 - 12:45, 14:00 - 17:00, April 15 - May 20, 2018
Room 440, Astronomy-Mathematics Building, NTU
Instructor:
Wei-Ming Ni (University of Minnesota)
Jong-Shenq Guo (Tamkang University)
Chang-Hong Wu (National University of Tainan)
Hirokazu Ninomiya (Meiji University)
Jann-Long Chern (National Central University)
Tai-Chia Lin (National Taiwan University)
Chao-Nien Chen (National Tsing Hua University)
Chiun-Chuan Chen (National Taiwan University)
Sze-Bi Hsu (National Tsing Hua University)
Feng-Bin Wang (Chang Gung University)
Organizers:
Chiun-Chuan Chen (National Taiwan University)
Chun-Hsiung Hsia (National Taiwan University)
Background & Purpose：
In the past two decades, in mathematical biology, there have been shown plenty of scientifically interesting phenomena due the interaction between spatial heterogeneity and diffusion. 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, we invite 10 mathematicians ) who's research fields are close
related to the aforementioned issues to conduct a series of lectures that would cover quite a large range research interests of mathematical biology. It is our hope that this lecture series would cultivate both the research ability and the research taste of our young scholars and students.
Course Outline
Part I.
Speaker: Wei-Ming Ni (University of Minnesota)
Title: Population Dynamics in Ecology: From Analysis to Ecology, and Back
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.
Part II.
Speakers: Sze-Bi Hsu (National Tsing Hua University),
Feng-Bin Wang (Chang Gung University)
Title 1. On the mathematical models of the competition of wild type bacteria and its drug-resistant mutants
The increase in antibiotic resistance is a world- wide public health problem. There are numerous studies on the evolution of the antibiotic –resistance. In this three-hour mini- course we shall use ODE and PDE to introduce mathematical models to describe the dynamics of evolution of antibiotic resistance bacteria. Based on the results of mathematical analysis, we give some biological interpretations of the competition outcomes of the bacteria. Several open problems are proposed to challenge the students. The outlines of the three-hour mini-course are followings:
(1) Analysis of a model of two competitors in a chemostat with external inhibitors.
(2) Morbidostat: A bio-reactor that promotes selection for drug resistance in bacteria
(3) Spatiotemporal microbial evolution of antibiotic resistance in bacteria.
References:
[1] S.B. Hsu and P.E. Waltman, SIAM J. Applied Math. (1992), 52,528-40
[2] Z.Chen, S.B. Hsu, Y.T. Yang, SIAM J. Applied Math. (2017), 77, p.470-499
[3] Roy Kishony et., Science (2016), Sept. Vol. 353, Issue 6304, 1147-1151
Title 2. Differential equations modeling the competition for resources with internal storage in variable habitats
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.
Part III.
Speaker: Hirokazu Ninomiya (Meiji University)
Title: Entire solutions of reaction-diffusion equations with bi-stable nonlinearity
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.
Part IV.
Speakers: Jong-Shenq Guo (Tamkang University),
Chang-Hong Wu (National University of Tainan)
Title 1. Spatio-temporal oscillation for a singular predator-prey model
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.
Title 2. Dynamics for some reaction-diffusion-advection models with free boundary.
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.
Part V.
Speakers: Jann-Long Chern (National Central University),
Zhi-You Chen (National Changhua University of Education)
Title 1. Asymptotic behavior of equilibrium states of reaction–diffusion systems with mass conservation
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.
Part VI.
Speakers: Chao-Nien Chen (National Tsing Hua University)
Chiun-Chuan Chen (National Taiwan University)
Title: Activator-inhibitor models with skew-gradient structure
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.
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 Equations 264 (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.
Title 2. The symmetry of least-energy solutions for semilinear elliptic equations
Recently in the research area of nonlinear elliptic PDEs, there have been many works devoted to studying problems where solutions exhibit the ‘‘phenomenon of point-condensation’’. Two well-known examples are semilinear elliptic equations involving the Sobolev critical exponent and nonlinear elliptic equations with small diffusion coefficient. These works show that the concentration often induces the asymptotic symmetry. For example, spherical Harnack inequalities have been proved for blowup solutions to either mean field equations on compact Riemann surfaces or the scalar curvature equation. These spherical Harnack inequalities implies that blowup solutions usually are asymptotically symmetric. Similar results were proved for spike-layer solutions of singularly elliptic Neumann problem. See [CL1, CL2, L1, L2, NT1, NT2] for more precise statements. Naturally, when the underlying equation is invariant under a group of transformations, we would like to know whether solutions with point-condensation actually possess certain symmetry which is invariant under the action of some elements of the group. In [Ln1, Ln2], for the mean field equation on S2; the second author first succeeded to prove the axial symmetry for solutions with two blowup points. In this talk we will study the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main discussions is to consider the least-energy solutions of some elliptic equations under some reasonable condition. We will study the example of the general phenomenon of the symmetry induced by point-condensation. A fine estimate for least-energy solution is required for the proof of symmetry of solutions. Meanwhile, we will also discuss some interesting open problems in this talk.
References
[AR] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.
[BN] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. 22 (1991) 1–37.
[CGS] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth, Comm. Pure Appl. Math. 42 (1989) 271–297.
[CL1] C.-C. Chen, C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. L (1997) 0971–1017.
[CL2] C.-C. Chen, C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom. 49 (1998) 115–178.
[CL3] C.-C. Chern, C.-S. Lin, On axisymmetric solutions of the conformal scalar curvature equation on Sn; Adv. Differential Equations 805 (2000) 121–146.
[CL] J.-L. Chern and C.-S. Lin, The symmetry of least-energy solutions for semilinear elliptic equations. J. Differential Equations 187 (2003), no. 2, 240–268.
[GNN1] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243.
[GNN2] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear equation in Rn; in: Mathematical Analysis and Applications, Part A, Academic Press, New York, Adv. Math. Suppl. Stud. 7A (1981) 369–402.
[H] Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincare’ Anal. Nonline’aire 8 (1991) 159–174.
[HL] Q. Han, F.-H. Lin, Elliptic Partial Differential Equations, Courant Institute of Mathematical Sciences, New York University.
[KS] E.F. Keller, L.A. Segal, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415.
[L1] Y.Y. Li, Harnack type inequality through the method of moving planes, Comm. Math. Phys. 2001 (1999) 421–444.
[L2] Y.Y. Li, Prescribing scalar curvature on Sn and related problems, Part I, J. Differential Equations 120 (1996) 541–597.
[Ln1] C.-S. Lin, Topological degree for a mean field equation on S2; Duke Math. J. 104 (2000) 5-1-536.
[Ln2] C.S. Lin, Locating the peaks of solutions via the maximum principle, I. Neumann problem, Comm. Pure Appl. Math. 54 (2001) 1065–1095.
[LN] C.S. Lin, W.M. Ni, On the diffusion coefficient of a semilinear Neumann problem, in: S. Hildebrandt, D. Kinderleher, M. Mirandar (Eds.), Lecture Notes in Math. Vol. 1340, pp. 160–174, Springer-Verlag, Berlin, New York.
[LNT] C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988) 1–27.
[NT1] W.-M. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. XLIV (1991) 819–851.
[NT2] W.-M. Ni, I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247–281.