Stochastics and Finance Seminar: Alexandru Hening -- Stochastic population dynamics

Dr Alexandru Hening from tufts University will deliver two seminars on stochastic
population dynamics.  

Seminar 1: 

TITLE.  Harvesting of populations in stochastic environments 

ABSTRACT.  We consider the harvesting of a population in a stochastic environment whose
dynamics in the absence of harvesting is described by a one dimensional diffusion.
Using ergodic optimal control, we find the optimal harvesting strategy which maximizes
the asymptotic yield of harvested individuals.  When the yield function is the identity,
we show that the optimal strategy has a bang-bang property: there exists a threshold
x*>0 such that whenever the population is under the threshold the harvesting rate
must be zero, whereas when the population is above the threshold the harvesting rate
must be at the upper limit.  We provide upper and lower bounds on the maximal asymptotic
yield, and explore via numerical simulations how the harvesting threshold and the
maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the
competition rate.  We also show that, if the yield function is 𝐶2 and strictly
concave, then the optimal harvesting strategy is continuous, whereas when the yield
function is convex the optimal strategy is of bang-bang type.  This shows that one
cannot always expect bang-bang type optimal controls.  

SEMINar 2: 

TITLE.  The competitive exclusion principle in stochastic environments 

ABSTRACT.  The competitive exclusion principle states that a number of species competing
for a smaller number of resources cannot coexist.  Even though this is a fundamental
principle in ecology, it has been observed empirically that in some settings it will
fail.  One example is Hutchinson’s ` paradox of the plankton’.  This is an instance
where a large number of phytoplankton species coexist while competing for a very limited
number of resources.  Both experimental and theoretical studies have shown that in some
instances (deterministic) temporal fluctuations of the environment can facilitate
coexistence for competing species.  Hutchinson conjectured that one can get coexistence
because nonequilibrium conditions would make it possible for different species to be
favored by the environment at different times.  In this talk I will look at how
environmental noise interacts with competitive exclusion.  I will show that, contrary to
Hutchinson’s explanation, one can switch between two environments in which the same
species is favored and still get coexistence.