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In many biological models it is necessary to allow the rates of change of the variables to depend on the past history, rather than only the current values, of the variables. The models may require discrete lags, with the use of delay-differential equations, or distributed lags, with the use of integro-differential equations. In these lecture notes I discuss the reasons for including lags, especially distributed lags, in biological models. These reasons may be inherent in the system studied, or may be the result of simplifying assumptions made in the model used. I examine some of the techniques available for studying the solution of the equations. A large proportion of the material presented relates to a special method that can be applied to a particular class of distributed lags. This method uses an extended set of ordinary differential equations. I examine the local stability of equilibrium points, and the existence and frequency of periodic solutions. I discuss the qualitative effects of lags, and how these differ according to the choice of discrete or distributed lag. The models studied are drawn from the population dynamiCS of single species (logistic growth, the chemostat) and of interacting pairs of species (predation, mutualism), from cell population dynamiCS (haemopoiesis) and from biochemical kinetics (the Goodwin oscillator). The last chapter is devoted to a population model employing difference equations. All these models include non-linear terms.
Hardcover:
9781771380478 | Kids Can Pr, April 1, 2015, cover price $18.95
Paperback:
9780387090924, titled "Time Lags in Biological Models" | Springer Verlag, June 1, 1979, cover price $28.00 | also contains Time Lags in Biological Models | About this edition: In many biological models it is necessary to allow the rates of change of the variables to depend on the past history, rather than only the current values, of the variables.
Paperback:
9781482418026 | Gareth Stevens Pub, January 1, 2015, cover price $8.15
9781482418002 | Gareth Stevens Pub, January 1, 2015, cover price $48.90
Library:
9781482418019 | Gareth Stevens Pub, January 1, 2015, cover price $22.60
Library:
9781482423600, titled "Mi dÃÂa en la escuela / My Day at School" | Bilingual edition (Gareth Stevens Pub, January 1, 2015), cover price $22.60
Paperback:
9781433990427 | Gareth Stevens Pub, August 1, 2013, cover price $8.15
Library:
9781433990410 | Gareth Stevens Pub, August 1, 2013, cover price $22.60
Paperback:
9781427109873 | Pap/com edition (Crabtree Pub Co, February 15, 2011), cover price $12.95 | also contains At School
CD/Spoken Word:
9781427109866 | Crabtree Pub Co, February 15, 2011, cover price $10.00 | also contains At School
Library:
9781427109880 | Lib/com edition (Crabtree Pub Co, February 15, 2011), cover price $25.60 | also contains At School
9780778794974 | 1 edition (Crabtree Pub Co, August 1, 2010), cover price $18.60 | also contains At School
Paperback:
9781427109873 | Pap/com edition (Crabtree Pub Co, February 15, 2011), cover price $12.95 | also contains At School
CD/Spoken Word:
9781427109866 | Crabtree Pub Co, February 15, 2011, cover price $10.00 | also contains At School
Library:
9781427109880 | Lib/com edition (Crabtree Pub Co, February 15, 2011), cover price $25.60 | also contains At School
9780778794974 | 1 edition (Crabtree Pub Co, August 1, 2010), cover price $18.60 | also contains At School
Product Description: In many biological models it is necessary to allow the rates of change of the variables to depend on the past history, rather than only the current values, of the variables. The models may require discrete lags, with the use of delay-differential equations, or distributed lags, with the use of integro-differential equations...read more
Paperback:
9780387090924 | Springer Verlag, June 1, 1979, cover price $28.00 | also contains School Days Around the World | About this edition: In many biological models it is necessary to allow the rates of change of the variables to depend on the past history, rather than only the current values, of the variables.
9783540090922 | Springer Verlag, December 18, 1978, cover price $129.00
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