2,048 research outputs found
Limit Cycles of Dynamic Systems under Random Perturbations with Rapid Switching and Slow Diffusion: A Multi-Scale Approach
This work is devoted to examining qualitative properties of dynamic systems,
in particular, limit cycles of stochastic differential equations with both
rapid switching and small diffusion. The systems are featured by multi-scale
formulation, highlighted by the presence of two small parameters and
.
Associated with the underlying systems, there are averaged or limit systems.
Suppose that for each pair of the parameters, the solution of the
corresponding equation has an invariant probability measure
, and that the averaged equation has a limit cycle in
which there is an averaged occupation measure for the averaged
equation. Our main effort is to prove that converges
weakly to as and under suitable
conditions. Moreover, our results are applied to a stochastic predator-prey
model together with numerical examples for demonstration.Comment: 25 page
Long-Run Average Sustainable Harvesting Policies: Near Optimality
This paper develops near-optimal sustainable harvesting strategies for the
predator in a predator-prey system. The objective function is of long-run
average per unit time type. To date, ecological systems under environmental
noise are usually modeled as stochastic differential equations driven by a
Brownian motion. Recognizing that the formulation using a Brownian motion is
only an idealization, in this paper, it is assumed that the environment is
subject to disturbances characterized by a jump process with rapid jump rates.
Under broad conditions, it is shown that the systems under consideration can be
approximated by a controlled diffusion system. Based on the limit diffusion
system, control policies of the original systems are constructed. Such an
approach enables us to develop sustainable harvesting policies leading to near
optimality. To treat the underlying problems, one of the main difficulties is
due to the long-run average objective function. This in turn, requires the
handling of a number of issues related to ergodicity. New approaches are
developed to obtain the tightness of the underlying processes based on the
population dynamic systems.Comment: 26 page
Stochastic Lotka-Volterra food chains
We study the persistence and extinction of species in a simple food chain
that is modelled by a Lotka-Volterra system with environmental stochasticity.
There exist sharp results for deterministic Lotka-Volterra systems in the
literature but few for their stochastic counterparts. The food chain we analyze
consists of one prey and predators. The th predator eats the th
species and is eaten by the th predator; this way each species only
interacts with at most two other species - the ones that are immediately above
or below it in the trophic chain. We show that one can classify, based on an
explicit quantity depending on the interaction coefficients of the system,
which species go extinct and which converge to their unique invariant
probability measure. Our work can be seen as a natural extension of the
deterministic results of Gard and Hallam '79 to a stochastic setting.
As one consequence we show that environmental stochasticity makes species
more likely to go extinct. However, if the environmental fluctuations are
small, persistence in the deterministic setting is preserved in the stochastic
system. Our analysis also shows that the addition of a new apex predator makes,
as expected, the different species more prone to extinction.
Another novelty of our analysis is the fact that we can describe the behavior
the system when the noise is degenerate. This is relevant because of the
possibility of strong correlations between the effects of the environment on
the different species.Comment: 22 page
The competitive exclusion principle in stochastic environments
In its simplest form, the competitive exclusion principle states that a
number of species competing for a smaller number of resources cannot coexist.
However, it has been observed empirically that in some settings it is possible
to have coexistence. 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 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 paper we show in various settings how a variable (stochastic)
environment enables a set of competing species limited by a smaller number of
resources or other density dependent factors to coexist. If the environmental
fluctuations are modeled by white noise, and the per-capita growth rates of the
competitors depend linearly on the resources, we prove that there is
competitive exclusion. However, if either the dependence between the growth
rates and the resources is not linear or the white noise term is nonlinear we
show that coexistence on fewer resources than species is possible. Even more
surprisingly, if the temporal environmental variation comes from switching the
environment at random times between a finite number of possible states, it is
possible for all species to coexist even if the growth rates depend linearly on
the resources. We show in an example (a variant of which first appeared in
Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can
switch between two environments in which the same species is favored and still
get coexistence.Comment: 26 pages, 4 figures, to appear in Journal of Mathematical Biolog
Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set
This work focuses on stability of regime-switching diffusions consisting of
continuous and discrete components, in which the discrete component switches in
a countably infinite set and its switching rates at current time depend on the
continuous component. In contrast to the existing approach, this work provides
more practically viable approach with more feasible conditions for stability. A
classical approach for asymptotic stabilityusing Lyapunov function techniques
shows the Lyapunov function evaluated at the solution process goes to 0 as time
. A distinctive feature of this paper is to obtain estimates of
path-wise rates of convergence, which pinpoints how fast the aforementioned
convergence to 0 taking place. Finally, some examples are given to illustrate
our findings.Comment: 25page
Coexistence and extinction for stochastic Kolmogorov systems
In recent years there has been a growing interest in the study of the
dynamics of stochastic populations. A key question in population biology is to
understand the conditions under which populations coexist or go extinct.
Theoretical and empirical studies have shown that coexistence can be
facilitated or negated by both biotic interactions and environmental
fluctuations. We study the dynamics of populations that live in a
stochastic environment and which can interact nonlinearly (through competition
for resources, predator-prey behavior, etc.). Our models are described by
-dimensional Kolmogorov systems with white noise (stochastic differential
equations - SDE). We give sharp conditions under which the populations converge
exponentially fast to their unique stationary distribution as well as
conditions under which some populations go extinct exponentially fast.
The analysis is done by a careful study of the properties of the invariant
measures of the process that are supported on the boundary of the domain. To
our knowledge this is one of the first general results describing the
asymptotic behavior of stochastic Kolmogorov systems in non-compact domains.
We are able to fully describe the properties of many of the SDE that appear
in the literature. In particular, we extend results on two dimensional
Lotka-Volterra models, two dimensional predator-prey models, dimensional
simple food chains, and two predator and one prey models. We also show how one
can use our methods to classify the dynamics of any two-dimensional stochastic
Kolmogorov system satisfying some mild assumptions.Comment: 37 page
Recurrence and Ergodicity of Switching Diffusions with Past-Dependent Switching Having A Countable State Space
This work focuses on recurrence and ergodicity of switching diffusions
consisting of continuous and discrete components, in which the discrete
component takes values in a countably infinite set and the rates of switching
at current time depend on the value of the continuous component over an
interval including certain past history. Sufficient conditions for recurrence
and ergodicity are given. Moreover, the relationship between systems of partial
differential equations and recurrence when the switching is past-independent is
established under suitable conditions.Comment: Potential Analysi
Persistence in Stochastic Lotka--Volterra food chains with intraspecific competition
This paper is devoted to the analysis of a simple Lotka-Volterra food chain
evolving in a stochastic environment. It can be seen as the companion paper of
Hening and Nguyen (J. of Math. Biol. `18) where we have characterized the
persistence and extinction of such a food chain under the assumption that there
is no intraspecific competition among predators. In the current paper we focus
on the case when all the species experience intracompetition. The food chain we
analyze consists of one prey and predators. The th predator eats the
st species and is eaten by the st predator; this way each species
only interacts with at most two other species - the ones that are immediately
above or below it in the trophic chain. We show that one can classify, based on
the invasion rates of the predators (which we can determine from the
interaction coefficients of the system via an algorithm), which species go
extinct and which converge to their unique invariant probability measure. We
obtain stronger results than in the case with no intraspecific competition
because in this setting we can make use of the general results of Hening and
Nguyen (Ann. of Appl. Probab.). Unlike most of the results available in the
literature, we provide an in depth analysis for both non-degenerate and
degenerate noise.
We exhibit our general results by analysing trophic cascades in a
plant--herbivore--predator system and providing persistence/extinction criteria
for food chains of length .Comment: 27 pages, 6 figures, corrected typos and added some applications to
trophic cascade
Modeling and Analysis of Switching Diffusion Systems: Past-Dependent Switching with a Countable State Space
Motivated by networked systems in random environment and controlled hybrid
stochastic dynamic systems, this work focuses on modeling and analysis of a
class of switching diffusions consisting of continuous and discrete components.
Novel features of the models include the discrete component taking values in a
countably infinite set, and the switching depending on the value of the
continuous component involving past history. In this work, the existence and
uniqueness of solutions of the associated stochastic differential equations are
obtained. In addition, Markov and Feller properties of a function-valued
stochastic process associated with the hybrid diffusion are also proved. In
particular, when the switching rates depend only on the current state, strong
Feller properties are obtained. These properties will pave a way for future
study of control design and optimization of such dynamic systems.Comment: We clarify a notation in Lemma A.1, which is not defined in the
journal versio
Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
We study the long-term qualitative behavior of randomly perturbed dynamical
systems. More specifically, we look at limit cycles of stochastic differential
equations (SDE) with Markovian switching, in which the process switches at
random times among different systems of SDEs, when the switching is fast and
the diffusion (white noise) term is small. The system is modeled by where is
a finite state space Markov chain with irreducible generator . The
relative changing rates of the switching and the diffusion are highlighted by
the two small parameters and . We associate to the system
the averaged ODE where and is the
unique invariant probability measure of the Markov chain with generator .
Suppose that for each pair of parameters, the process has
an invariant probability measure , and that the averaged
ODE has a limit cycle in which there is an averaged occupation measure
for the averaged equation. We are able to prove that if has finitely
many unstable or hyperbolic fixed points, then
converges weakly to as and . Our results
generalize to the setting of state-dependent switching as long as the
generator is bounded, Lipschitz, and irreducible for
all . We conclude our analysis by studying a predator-prey
model.Comment: 40 page
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