How do populations evolve? This question inspired Alberto Saldaña
to his PhD thesis on Partial symmetries of solutions to nonlinear
elliptic and parabolic problems in bounded radial domains. He
considered an extended Lotka-Volterra models which is describing
the dynamics of two species such as wolves in a bounded radial
domain:
(u1)t−μ1Δu1=a1(t)u1−b1(t)u12−α1(t)u1u2(u2)t−μ2Δu2=a2(t)u2−b2(t)u22−α2(t)u1u2
For each species, the model contains the diffusion μ of a
individual beings, the birth rate a, the saturation rate or
concentration b, and the aggressiveness rate α.
Starting from an initial condition, a distribution of u1 and u2
in the regarded domain, above equations with additional
constraints for well-posedness will describe the future outcome.
In the long run, this could either be co-existence, or extinction
of one or both species. In case of co-existence, the question is
how they will separate on the assumed radial bounded domain. For
this, he adapted a moving plane method.
On a bounded domain, the given boundary conditions are an
important aspect for the mathematical model: In this setup, a
homogeneous Neumann boundary condition can represent a fence,
which no-one, or no wolve, can cross, wereas a homogeneous
Dirichlet boundary condition assumes a lethal boundary, such as
an electric fence or cliff, which sets the density of living, or
surviving, individuals touching the boundary to zero.
The initial conditions, that is the distribution of the wolf
species, were quite general but assumed to be nearly reflectional
symmetric.
The analytical treatment of the system was less tedious in the
case of Neumann boundary conditions due to reflection symmetry at
the boundary, similar to the method of image charges in
electrostatics. The case of Dirichlet boundary conditions needed
more analytical results, such as the Serrin's boundary point
lemma. It turned out, that asymtotically in both cases the two
species will separate into two symmetric functions. Here, Saldaña
introduced a new aspect to this problem: He let the birth rate,
saturation rate and agressiveness rate vary in time. This
time-dependence modelled seasons, such as wolves behaviour
depends on food availability.
The Lotka-Volterra model can also be adapted to a predator-prey
setting or a cooperative setting, where the two species live
symbiotically. In the latter case, there also is an asymptotical
solution, in which the two species do not separate- they stay
together.
Alberto Saldaña startet his academic career in Mexico where he
found his love for mathematical analysis. He then did his Ph.D.
in Frankfurt, and now he is a Post-Doc in the Mathematical
Department at the University of Brussels.
Literature and additional material
A. Saldaña, T. Weth: On the asymptotic shape of solutions to
Neumann problems for non-cooperative parabolic systems, Journal
of Dynamics and Differential Equations,Volume 27, Issue 2, pp
307-332, 2015.
A. Saldaña: Qualitative properties of coexistence and
semi-trivial limit profiles of nonautonomous nonlinear parabolic
Dirichlet systems, Nonlinear Analysis: Theory, Methods and
Applications, 130:31 46, 2016.
A. Saldaña: Partial symmetries of solutions to nonlinear
elliptic and parabolic problems in bounded radial domains, PhD
thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main,
Germany, 2014.
A. Saldaña, T. Weth: Asymptotic axial symmetry of solutions
of parabolic equations in bounded radial domains, Journal of
Evolution Equations 12.3: 697-712, 2012.
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