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Chemosensitive movement describes the active orientation of individuals on chemical signals. In cases of cellular slime molds or flagellated bacteria, chemosensitive movement leads to aggregation and pattern formation. The classical mathematical model to describe chemosensitive movement is the diffusion based Patlak–Keller–Segel model. It suffers from the drawback of infinite propagation speeds. The relevant model parameters (motility and chemosensitivity) are related to population statistics. Hyperbolic models respect finite propagation speeds and the relevant model parameters (turning rate, distribution of new chosen velocities) are based on the individual movement patterns of the species at hand. In this paper hyperbolic models (in 1-D) and a transport model (in n-D) for chemosensitive movement are discussed and compared to the classical model. For the hyperbolic and transport models the following topics are reviewed: parabolic limit (which in some cases leads to the Patlak–Keller–Segel model), local and global existence, asymptotic behavior and moment closure. The moment closure approach leads to models based on Cattaneo's law of heat conduction (telegraph equation).

In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.

For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.

We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

In this paper we consider kinetic and associated macroscopic models for chemotaxis on a network. Coupling conditions at the nodes of the network for the kinetic problem are presented and used to derive coupling conditions for the macroscopic approximations. The results of the different models are compared and relations to a Keller–Segel model on networks are discussed. For a numerical approximation of the governing equations asymptotic preserving relaxation schemes are extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

Starting from a two-scale description involving receptor binding dynamics and a kinetic transport equation for the evolution of the cell density function under velocity reorientations, we deduce macroscopic models for glioma invasion featuring partial differential equations for the mass density and momentum of a population of glioma cells migrating through the anisotropic brain tissue. The proposed first and higher-order moment closure methods enable numerical simulations of the kinetic equation. Their performance is then compared to that of the diffusion limit. The approach allows for diffusion tensor imaging (DTI)-based, patient-specific predictions of the tumor extent and its dynamic behavior.

Starting from a nonlocal version of a classical kinetic traffic model, we derive a class of second-order macroscopic traffic flow models using appropriate moment closure approaches. Under mild assumptions on the closure, we prove that the resulting macroscopic equations fulfill a set of conditions including hyperbolicity, physically reasonable invariant domains and physically reasonable bounds on the speed with which the waves propagate. Finally, numerical results for various situations are presented, illustrating the analytical findings and comparing kinetic and macroscopic solutions.

Aphids, the sap-sucking insects, often feeding in clusters on new plant growth, have resulted in large amounts of resources and efforts being spent attempting to control their activities. Taking cotton aphids as an example, this paper presents optimal control problems governed by stochastic models with impulsive interferences. Differing from the moment closure equation methods which are computationally intractable when the model contains excessive species, a new computational approach is employed to solve this problem. The key of the approach is to establish a functional relationship between the control variables involving the releasing rates of sterile insects and the spraying rates of pesticide and corresponding states on aphids and sterile aphids. Then the log-linear regression model is proposed to link the control variables with the moments (including the mean and variance) of states. Using training sample simulated from Gillespie algorithm, the regression coefficients for constraints and the objective function are estimated by least squares method. Simulation shows the error of the prediction of this model is relatively low and control results based on regression model are superior to the method based on the moment closure equations in terms of the control cost. Finally, the relative impacts of the prices and area of the field on optimal tactics are explored.

Two stochastic models are derived for a susceptible–infectious–susceptible epidemic spreading through a metapopulation: a continuous time Markov chain (CTMC) model and an Itô stochastic differential equation (SDE) model. The stochastic models are numerically compared. Close agreement suggests that computationally intense CTMC simulations can be approximated by simpler SDE simulations. Differential equations for the moments of the SDE probability distribution are also derived, the steady states are solved numerically using a moment closure technique, and these results are compared to simulations. The moment closure technique only coarsely approximates simulation results. The effect of model parameters on stability of the disease-free equilibrium is also numerically investigated.

A limitation of most whole cell models to date is the assumption that intracellular Ca²⁺ channels are globally coupled by a continuously stirred bulk cytosolic [Ca²⁺], when in fact open intracellular Ca²⁺ channels experience elevated domain [Ca²⁺]. Such heterogeneous local [Ca²⁺] can be modeled using a 2N+2-compartment model that includes bulk cytosolic and luminal [Ca²⁺] and 2N compartments representing cytosolic and luminal Ca²⁺ domains associated with N stochastically gating intracellular channels. We have introduced an alternative whole cell model formulation that solves a system of advection-reaction equations for the probability density of cytosolic and luminal domain [Ca²⁺] jointly distributed with channel state. This probability density formulation and an associated moment closure approach have been used to create computationally efficient models of local control of Ca²⁺-induced Ca²⁺ release in ventricular cardiac myocytes.