# Simple mathematical models with very complicated dynamics

@article{May1976SimpleMM, title={Simple mathematical models with very complicated dynamics}, author={Robert M. May}, journal={Nature}, year={1976}, volume={261}, pages={459-467} }

First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practicalâ€¦Â

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## References

SHOWING 1-10 OF 69 REFERENCES

Deterministic nonperiodic flow

- Mathematics
- 1963

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified withâ€¦

Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos

- Mathematics, MedicineScience
- 1974

This paper presents a dynamical regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur.

Nonlinear Aspects of Competition Between Three Species

- Mathematics
- 1975

It is shown that for three competitors, the classic Gauseâ€“Lotkaâ€“Volterra equations possess a special class of periodic limit cycle solutions, and a general class of solutions in which the systemâ€¦

Biological populations obeying difference equations: stable points, stable cycles, and chaos.

- Mathematics, MedicineJournal of theoretical biology
- 1975

The corresponding simplest difference equations, with their built-in time lag in the operation of regulatory mechanisms, can have a complicated dynamical structure, the great richness of which is not commonly appreciated either in the ecological literature, or in elementary mathematical analysis.

Dynamic complexity in predator-prey models framed in difference equations

- MathematicsNature
- 1975

THE complicated dynamics associated with simple first-order, nonlinear difference equations have received considerable attention (refs 1â€“4 and R. M. May and G. F. Oster, unpublished). In anâ€¦

Period Three Implies Chaos

- Mathematics
- 1975

The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenonâ€¦

Sensitivity problems related to certain bifurcations in non-linear recurrence relations

- Mathematics, Computer ScienceAutom.
- 1969

This paper is concerned with certain qualitative aspects of the sensitivity problem in relation to small variations of a parameter of a system, the behaviour of which can be described by anâ€¦

Some remarks on animal population dynamics.

- MedicineBiometrics
- 1950

T HE PRESENT PAPER is not a survey of the vast field of investigation into the dynamics of population change but an attempt to discuss a few specific problems in the hope of suggesting some new linesâ€¦

Oscillation in the Simple Logistic Growth Model

- MathematicsNature
- 1965

THE logistic curve is often used in teaching ecology as a first description of growth of an animal population. For many reasons, frequency related to age structure and time-lag effects, it does notâ€¦