Individual-based models, 'IBMs', describe naturally the dynamics of interacting organisms or social or financial agents. They are considered too complex for mathematical analysis, but computer ...simulations of them cannot give the general insights required. Here, we resolve this problem with a general mathematical framework for IBMs containing interactions of an unlimited level of complexity, and derive equations that reliably approximate the effects of space and stochasticity. We provide software, specified in an accessible and intuitive graphical way, so any researcher can obtain analytical and simulation results for any particular IBM without algebraic manipulation. We illustrate the framework with examples from movement ecology, conservation biology, and evolutionary ecology. This framework will provide unprecedented insights into a hitherto intractable panoply of complex models across many scientific fields.
We describe a general approach to the construction of a state evolution corresponding to the Markov generator of a spatial birth-and-death dynamics in
R
d
. We present conditions on the ...birth-and-death intensities which are sufficient for the existence of an evolution as a strongly continuous semigroup in a proper Banach space of correlation functions satisfying the Ruelle bound. The convergence of a Vlasov-type scaling for the corresponding stochastic dynamics is considered.
We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on R^d. Using ...the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.
We consider the extinction regime in the spatial stochastic logistic model in
(a.k.a. Bolker-Pacala-Dieckmann-Law model of spatial populations) using the first-order perturbation beyond the ...mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality - the smallest constant death rate which ensures the extinction of the population - as a function of the mean-field scaling parameter
. We find the leading term of the asymptotic expansion (as
) of the critical mortality which is apparently different for the cases
, d = 2, and d = 1.
Spatial and stochastic models are often straightforward to simulate but difficult to analyze mathematically. Most of the mathematical methods available for nonlinear stochastic and spatial models are ...based on heuristic rather than mathematically justified assumptions, so that, e.g., the choice of the moment closure can be considered more of an art than a science. In this paper, we build on recent developments in specific branch of probability theory, Markov evolutions in the space of locally finite configurations, to develop a mathematically rigorous and practical framework that we expect to be widely applicable for theoretical ecology. In particular, we show how spatial moment equations of all orders can be systematically derived from the underlying individual-based assumptions. Further, as a new mathematical development, we go beyond mean-field theory by discussing how spatial moment equations can be perturbatively expanded around the mean-field model. While we have suggested such a perturbation expansion in our previous research, the present paper gives a rigorous mathematical justification. In addition to bringing mathematical rigor, the application of the mathematically well-established framework of Markov evolutions allows one to derive perturbation expansions in a transparent and systematic manner, which we hope will facilitate the application of the methods in theoretical ecology.
We consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at ...0), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both
, perhaps different. We show that, in such case, the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early.
We consider an infinite system of first order differential equations in
R
ν
, parameterized by elements
x
of a fixed countable set
γ
⊂
R
d
, where the right-hand side of each
x
-equation depends on a ...finite but in general unbounded number
n
x
of variables (a row-finite system). Such systems describe in particular (non-equilibrium) dynamics of spins
q
x
∈
R
ν
of a collection of particles labelled by points
x
∈
γ
. Two spins
q
x
and
q
y
interact via a pair potential if the distance between
x
and
y
is no more than a fixed interaction radius. In contrast to the case where
γ
is a regular graph, e.g.
Z
d
, the number
n
x
of particles interacting with particle
x
can be unbounded in
x
. Our main example of a “growing” configuration
γ
is a typical realization of a Poisson (or Gibbs) point process. Under certain dissipativity-type condition on the right-hand side of our system and a bound on growth of
n
x
, we prove the existence and (under additional assumptions) uniqueness of infinite lifetime solutions with explicit estimates of growth in parameter
x
and time
t
. For this, we obtain uniform estimates of solutions to approximating finite systems using a version of Ovsyannikov’s method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.
We study traveling waves for a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions. We describe relations between ...speeds and asymptotic of profiles of traveling waves, and prove the uniqueness of the profiles up to shifts.
Agent-based models are used to study complex phenomena in many fields of science. While simulating agent-based models is often straightforward, predicting their behaviour mathematically has remained ...a key challenge. Recently developed mathematical methods allow the prediction of the emerging spatial patterns for a general class of agent-based models, whereas the prediction of spatio-temporal pattern has been thus far achieved only for special cases. We present a general and mathematically rigorous methodology that allows deriving the spatio-temporal correlation structure for a general class of individual-based models. To do so, we define an auxiliary model, in which each agent type of the primary model expands to three types, called the original, the past and the new agents. In this way, the auxiliary model keeps track of both the initial and current state of the primary model, and hence the spatio-temporal correlations of the primary model can be derived from the spatial correlations of the auxiliary model. We illustrate the agreement between analytical predictions and agent-based simulations using two example models from theoretical ecology. In particular, we show that the methodology is able to correctly predict the dynamical behaviour of a host-parasite model that shows spatially localized oscillations.
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