Effect of Time Delay and Age Structure

The inclusion of age structure in RD equations has its origin in the generalization of population growth models. Age-structured models take explicitly into account that population growth is due only to adult individuals. The oldest such model is described by the McKendrick-von Foerster equation [447]  

Here p, a, t) is the per capita mortality rate for individuals of age a at time t, and J(a, t) is the flux of individuals of age a at time t. Dividing by Aa and taking the limit Aa 0, we obtain the conservation equation for the density of individuals  

The flux J is not a flux in space, but rather the movement of individuals in age. We assume that it is proportional to the density of individuals and some characteristic velocity of aging, J a, t) = p a, t)v(a, t). Aging is simply the passage of time V = da/dt =, and we obtain (4.107). If we also include the flux in the space due to the motion of individuals, then we obtain Metz-Diekman model [295]  

Almost parallel to McKendrick, Hutchinson [215], a well-known ecologist, proposed a time-delayed version for the logistic growth equation, where the nonlinear term was delayed in time. The diffusive Hutchinson equation, also known as the delayed Fisher equation. 

Under certain conditions, this equation has front solutions. However, as for (4.111), loss of monotonicity occurs as the delay is increased and the front develops a prominent hump [167]. 

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