Journal of Mathematical Biology

Beretta, Edoardo; Solimano, Fortunata; Takeuchi, Yasuhiro

doi: 10.1007/s002850050039pmid: 9002239

 A mathematical model for the delivery of drug directly to the macrophages by using the phagocytosis of senescent red blood cells is proposed. The model is based on the following assumption: At time t=0 a preassigned red blood cell population n(0, a)=φ(a), a>0, loaded by the drug, is injected in the blood circulation. Among the cells of that population only those with an age a≧ā (ā=120 days) will be phagocytosed by macrophages. Of course, the lifetime of the drug must be higher than ā. Within the red blood cells it cannot be metabolized, neither can it diffuse through their membranes. The emphasis of the paper is on the mathematical properties and on the formulation of the control problem.

Lizana, Marcos; Rivero, Jesus

doi: 10.1007/s002850050040pmid: 9002240

 In the present paper we make a bifurcation analysis of an SIRS epidemiological model depending on all parameters. In particular we are interested in codimension-2 bifurcations.

Schreiber, Sebastian J.

doi: 10.1007/s002850050041pmid: N/A

 Models of population growth in consumer-resource cascades (serially arranged containers with a dynamic consumer population, v, receiving a flow of resource, u, from the previous container) with a functional response of the form h(u/v b ) are investigated. For b∈[0, 1], it is shown that these models have a globally stable equilibrium. As a result, two conclusions can be drawn: (1) Consumer density dependence in the functional or in the per-capita numerical response can result in persistence of the consumer population in all containers. (2) In the absence of consumer density dependence, the consumer goes extinct in all containers except possibly the first. Several variations of this model are discussed including replacing discrete containers by a spatial continuum and introducing a dynamic resource.

Hillen, Thomas

doi: 10.1007/s002850050042pmid: N/A

 If in the classical Turing model the diffusion process (Brownian motion) is replaced by a more general correlated random walk, then the parameters describing spatial spread are the particle speeds and the rates of change in direction. As in the Turing model, a spatially constant equilibrium can become unstable if the different species have different turning rates and different speeds. Furthermore, a Hopf bifurcation can be found if the reproduction rate of the activator is greater than its rate of change of direction, and oscillating patterns are possible.

Watanabe, Mari; Gilmour Jr., Robert F.

doi: 10.1007/s002850050043pmid: 9002241

 Heart rate-dependent alterations in the duration of the electrically active state of cardiac cells, the action potential, are an important determinant of lethal heart rhythm disorders. The relationship between action potential duration and heart rate can be modelled as a nonlinear one-dimensional map. Iteration of the map over a range of physiologically relevant heart rates produces complex changes in action potential duration, including period doubling bifurcations, chaos and period doubling reversals. We present a computer algorithm that ensures, over the same range of heart rates, uniform state variable values (action potential durations) by application of small perturbing stimuli at appropriate intervals. The algorithm succeeds, even though the only parameter in the system (the heart rate) is immutable. Control of the dynamics is achieved by exploiting the inexcitability of the cardiac cells immediately after stimulation. This algorithm may have applications for the prevention of cardiac rhythm disturbances.

Gimelfarb, A.

doi: 10.1007/s002850050044pmid: 9026554

 The existence of two stable, symmetric (allelic frequency 0.5 in each locus) polymorphic states is demonstrated for a two-locus model of an additive quantitative trait under strong Gaussian selection. Linkage disequilibrium at one of the states is negative whereas it is positive at the other state. For a three-locus model, it is shown that in order to maintain a stable polymorphism in all three loci, selection must be sufficiently but not exces- sively strong relative to recombination. Also, positive linkage disequilibrium can be maintained in a three-locus model under stabilizing selection that is not very strong.

Schwarz, Richard I.

doi: 10.1007/s002850050045pmid: 9002242

 A mathematical model of tendon morphogenesis is presented that is consistent with the dramatic transitions seen in this tissue as it progresses from rapid growth early in development to no growth in the adult. To accomplish this change, the embryonic chick tendon is hypercellular with each cell dedicating half of its protein production to procollagen but over time, as growth subsides, the tissue gradually becomes hypocellular with each cell producing only about 1% procollagen. Making this transition from the embryonic to the adult state, forming a roughly cylindrical tissue composed of ∼90% collagen, and linking the correct muscle to the right bone, is a complex task. The proposed solution requires only two factors: an activator of growth and an inhibitor complex, composed of the activator and another molecule that modifies the activity of the activator. From a diverse set of cell culture observations, these two factors were deduced as the primary components of the mechanism that allows cells to signal their presence to their neighbors. Since cell density signaling is the principal regulator of both collagen synthesis and cell proliferation, its components should play the key role in tendon development. A mathematical model based on the changes in the concentrations of these factors with cell density correlates well with the transitions observed in vivo. Furthermore, the model predicts that in the maturing chicken there should be a high cell density region at the muscle tendon interface. Experimental observations of frozen sections of tendon from a 4 month old chicken confirm this prediction.

Soto-Treviño, C.; Kopell, N.; Watson, D.

doi: 10.1007/s002850050046pmid: 9002243

 Many excitable membrane systems display bursting oscillations, in which the membrane potential switches periodically between an active phase of rapid spiking and a silent phase of slow, quasi steady-state behavior. A burster is called parabolic when the spike frequency is lower both at the beginning and end of the active phase. We show that classes of voltage-gated conductance equations can be reduced to the mathematical mechanism previously analyzed by Ermentrout and Kopell in [7]. The reduction uses a series of coordinate changes and shows that the mechanism in [7] applies more generally than previously believed. The key hypothesis for the more general theory is that a certain slow periodic orbit must stay close to a curve of degenerate homoclinic points for the fast system, at least during the active phase. We do not require that the slow system have a periodic orbit when the voltage is held constant.