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Construction of population growth equations in the presence of viability constraints

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Abstract

A mathematical method based on the G-projection of differential inclusions is used to construct dynamical models of population biology. We suppose that the system under study, not being limited by resources, may be described by a control system

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

\] where u is a control describing the choice of resources. Then considering the constraints that the system must satisfy we define a viability set K. Since there may not exist a control u(·) such that the corresponding solution satisfies x(t) ε K, we have to change the dynamics of the control system to get a viable solution. Using the G-projection we introduce so-called “projected” control system

$$\dot x\left( t \right) = \Pi _{T_K }^G f\left( {x\left( t \right),u\left( t \right)} \right)$$

that has a viable solution. The projected system has usually simpler dynamics than traditional models used in population biology.

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References

  1. Aubin, J.-P., Cellina, A.: Differential inclusions. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  2. Aubin, J.-P., Frankowska, H.: Set-valued analysis. Boston: Birkhaüser 1990

    Google Scholar 

  3. Henry, C.: Differential equations with discontinuous right hand side for planning procedures. J. Econ. Theory 4, 544–551 (1973)

    Google Scholar 

  4. Aubin, J.-P.: A survey of viability theory. SIAM J. Control and Optimization 28, 749–788 (1990)

    Google Scholar 

  5. Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Analysis Appl. 41, 179–186 (1973)

    Google Scholar 

  6. Křivan, V.: G-projection of differential inclusions. Preprint

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Křivan, V. Construction of population growth equations in the presence of viability constraints. J. Math. Biol. 29, 379–387 (1991). https://doi.org/10.1007/BF00167158

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  • DOI: https://doi.org/10.1007/BF00167158

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