29 citations · 73 across the 14 of their papers we have counts for
15 papers
Pattern Formation in Mesic Savannas
Denis D. Patterson, Simon A. Levin, A. Carla Staver +1
We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study…
Spatial Dynamics with Heterogeneity
Denis D. Patterson, Simon A. Levin, A. Carla Staver +1
Spatial systems with heterogeneities are ubiquitous in nature, from precipitation, temperature and soil gradients controlling vegetation growth to morphogen gradients controlling g…
Modeling Immunity to Malaria with an Age-Structured PDE Framework
Zhuolin Qu, Denis Patterson, Lauren Childs +4
Malaria is one of the deadliest infectious diseases globally, causing hundreds of thousands of deaths each year. It disproportionately affects young children, with two-thirds of fa…
Unifying deterministic and stochastic ecological dynamics via a landscape-flux approach
Li Xu, Denis Patterson, Ann Carla Staver +2
We develop a landscape-flux framework to investigate observed frequency distributions of vegetation and the stability of these ecological systems under fluctuations. The frequency…
Probabilistic Foundations of Spatial Mean-field Models in Ecology and Applications
Denis D. Patterson, Simon A. Levin, A. Carla Staver +1
Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between t…
Blow-up and superexponential growth in superlinear Volterra equations
John A. D. Appleby, Denis D. Patterson
This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp e…
Growth Rates of Solutions of Superlinear Ordinary Differential Equations
John A. D. Appleby, Denis D. Patterson
In this letter we obtain sharp estimates on the growth rate of solutions to a nonlinear ODE with a nonautonomous forcing term. The equation is superlinear in the state variable and…
Growth and fluctuation in perturbed nonlinear Volterra equations
John A. D. Appleby, Denis D. Patterson
We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces…
Hartman-Wintner growth results for sublinear functional differential equations
John A. D. Appleby, Denis D. Patterson
In this paper, we determine rates of growth to infinity of scalar autonomous nonlinear functional and Volterra differential equations. In these equations, the right-hand side is a…
Large fluctuations and growth rates of linear Volterra summation equations
John A. D. Appleby, Denis D. Patterson
This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel i…
On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations
John A. D. Appleby, Denis D. Patterson
In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state…
Memory Dependent Growth in Sublinear Volterra Differential Equations
John A. D. Appleby, Denis D. Patterson
We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. To obtain precise results we utilise the powerful theory of regular vari…
Growth Rates of Sublinear Functional and Volterra Differential Equations
John A. D. Appleby, Denis D. Patterson
This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymp…
Subexponential Growth Rates in Functional Differential Equations
John A. D. Appleby, Denis D. Patterson
This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive co…
On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity
John A. D. Appleby, Denis D. Patterson
This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential…