Differential Equations and Their Applications pp 372-475 | Cite as

# Qualitative theory of differential equations

Chapter

First Online:

## Abstract

In this chapter we consider the differential equation
where
and
is a nonlinear function of

$$\dot x = f\left( {t,x} \right)$$

(1)

$$
x = \left[ {\begin{array}{*{20}{c}}
{{x_1}\left( t \right)} \\
\vdots\\
{{x_n}\left( t \right)}
\end{array}} \right],
$$

$$
f\left( {t,x} \right) = \left[ {\begin{array}{*{20}{c}}
{{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \\
\vdots\\
{{f_n}\left( {t,{x_1}, \ldots ,{x_n}} \right)}
\end{array}} \right]
$$

*x*_{1}, ...,*x*_{n}. Unfortunately, there are no known methods of solving Equation. This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of explicitly. For example, let*x*_{1}(*t*) and*x*_{2}(*t*) denote the populations, at time*t*, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of*x*_{1}(*t*) and*x*_{2}(*t*) are governed by the differential equation. In this case, we are not really interested in the values of*x*_{1}(*t*) and*x*_{2}(*t*) at every time*t*. Rather, we are interested in the qualitative properties of*x*_{1}(*t*) and*x*_{2}(*t*). Specically, we wish to answer the following questions.## Keywords

Equilibrium Point Phase Portrait Bifurcation Point Equilibrium Solution Future Time
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Reference

- Richardson, L. F. F., “Generalized foreign politics,” The British Journal of Psychology, monograph supplement #23, 1939.Google Scholar
- 1.Coleman, C. S., Combat Models, MAA Workshop on Modules in Applied Math, Cornell University, Aug. 1976.Google Scholar
- 2.Engel, J. H., A verification of Lanchester’s law, Operations Research, 2, (1954), 163 - 171.Google Scholar
- 3.Howes, D. R., and Thrall, R. M., A theory of ideal linear weights for heteroge?neous combat forces, Naval Research Logistics Quarterly, vol. 20, 1973, pp. 645 - 659.CrossRefGoogle Scholar
- 4.Lanchester, F. W., Aircraft in Warfare, the Dawn of the Fourth Arm. Tiptree, Constable and Co., Ltd., 1916.zbMATHGoogle Scholar
- 5.Morehouse, C. P., The Iwo Jima Operation, USMCR, Historical Division, Hdqr. USMC, 1946.Google Scholar
- 6.Newcomb, R. F., Iwo Jima. New York: Holt, Rinehart, and Winston, 1965.Google Scholar
- Volterra, V: “Leons sur la theorie mathematique de la lutte pour la vie.” Paris, 1931.Google Scholar
- Gause, G. F. F., ‘The Struggle for Existence,’ Dover Publications, New York, 1964.Google Scholar
- Bailey, N. T. J., ‘The mathematical theory of epidemics,’ 1957, New York.Google Scholar
- Kermack, W. O. and McKendrick, A. G., Contributions to the mathematical therory of epidemics, Proceedings Roy. Stat. Soc., A, 115, 700 - 721, 1927.zbMATHGoogle Scholar
- Waltman, P, P., ‘Deterministic threshold models in the theory of epidemics,’ Springer-Verlag, New York, 1974.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1993