Multirate extrapolation methods for differential equations with different time scales

The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.

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Problems with Different Time Scales for Ordinary Differential Equations

SIAM Journal on Numerical Analysis ◽

10.1137/0716072 ◽

1979 ◽

Vol 16 (6) ◽

pp. 980-998 ◽

Cited By ~ 67

Author(s):

Heinz-Otto Kreiss

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Time Scales ◽

Different Time Scales

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Problems with different time scales for partial differential equations

Communications on Pure and Applied Mathematics ◽

10.1002/cpa.3160330310 ◽

1980 ◽

Vol 33 (3) ◽

pp. 399-439 ◽

Cited By ~ 121

Author(s):

Heinz-Otto Kreiss

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Time Scales ◽

Partial Differential ◽

Different Time Scales

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Agent-Based Computational Model

10.23943/princeton/9780691158884.003.0003 ◽

2017 ◽

Cited By ~ 2

Author(s):

Joshua M. Epstein

Keyword(s):

Computational Model ◽

Bounded Rationality ◽

Time Scales ◽

Spatially Explicit ◽

Agent Based ◽

Affective Component ◽

Elementary Type ◽

Social Component ◽

Different Time Scales

This part describes the agent-based and computational model for Agent_Zero and demonstrates its capacity for generative minimalism. It first explains the replicability of the model before offering an interpretation of the model by imagining a guerilla war like Vietnam, Afghanistan, or Iraq, where events transpire on a 2-D population of contiguous yellow patches. Each patch is occupied by a single stationary indigenous agent, which has two possible states: inactive and active. The discussion then turns to Agent_Zero's affective component and an elementary type of bounded rationality, as well as its social component, with particular emphasis on disposition, action, and pseudocode. Computational parables are then presented, including a parable relating to the slaughter of innocents through dispositional contagion. This part also shows how the model can capture three spatially explicit examples in which affect and probability change on different time scales.

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