Exact solutions of Lotka-Volterra equations

The Lotka-Volterra nonlinear differential equations for two competing species P and Q contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the P, Q phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated.     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Exact solutions to a spatially extended model of kinase-receptor interaction

B and Mast cells are activated by the aggregation of the immune receptors. Motivated by this phenomena we consider a simple spatially extended model of mutual interaction of kinases and membrane receptors. It is assumed that kinase activates membrane receptors and in turn the kinase molecules bound to the active receptors are activated by transphosphorylation. Such a type of interaction implies positive feedback and may lead to bistability. In this study we apply the Steklov eigenproblem theory to analyze the linearized model and find exact solutions in the case of non-uniformly distributed membrane receptors. This approach allows us to determine the critical value of receptor dephosphorylation rate at which cell activation (by arbitrary small perturbation of the inactive state) is possible. We found that cell sensitivity grows with decreasing kinase diffusion and increasing anisotropy of the receptor distribution. Moreover, these two effects are cooperating. We showed that the cell activity can be abruptly triggered by the formation of the receptor aggregate. Since the considered activation mechanism is not based on receptor crosslinking by polyvalent antigens, the proposed model can also explain B cell activation due to receptor aggregation following binding of monovalent antigens presented on the antigen presenting cell.     

Bifurcation,stability diagrams,and varying diffusion coefficients in reaction-diffusion equations

The effect of keeping all the parameters constant, except the diffusion coefficients, in a pair of reaction-diffusion equations is studied. It is shown that the stability of the constant solution and the bifurcation points can be easily established by constructing a simple stability diagram. The possible qualitatively different diagrams are enumerated.     

Exact inbreeding coefficients in populations with overlapping generations

A theory is given that allows inbreeding coefficients to be calculated exactly for populations with overlapping generations. Emphasis is placed on providing equations well suited for computer iteration. Both monoecious and dioecious populations are considered and family size is not restricted to being Poisson. One-locus and two-locus inbreeding coefficients are evaluated, although the reader may omit the two-locus sections. The exact treatment is shown to be preferable to approximate treatments in that it applies to both early and late generations for all populations sizes. Inbreeding effective numbers found by the exact treatment are compared to various approximate numbers, and the approximate values are found to be generally very good.     

Virial coefficients of protein solutions

An osmometer capable of measuring protein osmotic pressures up to 100 cms. of mercury pressure has been described. The principle of the osmometer is to set a given pressure and to permit the protein concentration to equilibrate with the pressure. The higher virial osmotic coefficients of egg albumin in various electrolytes and in 1 m urea as well as a function of NaCl concentration are reported. The virial coefficients of bovine serum albumin and of bovine methemoglobin in 1 m NaCl are also given. It appears that the primary cause for the departure of the osmotic pressure from ideality is due to the covolumes of the proteins.     

Comparing solutions to the linear and nonlinear CFK equations for predicting COHb formation

The Coburn, Forster, Kane model of COHb formation as a result of exposure to CO exists in two forms: the linear CFK equation that assumes a constant level of oxyhemoglobin; and the nonlinear CFK equation that allows the oxyhemoglobin level to vary with the carboxyhemoglobin level. Although both equations are currently used, no rigorous analysis exists to show under what conditions the two models determine substantially different solutions. This paper provides such an analysis and shows that the linear model may be used as a reasonable approximation over a much wider range of carboxyhemoglobin levels than had been supposed on physiological grounds.     

Exact digital simulation of time-invariant linear systems with applications to neuronal modeling

An efficient new method for the exact digital simulation of time-invariant linear systems is presented. Such systems are frequently encountered as models for neuronal systems, or as submodules of such systems. The matrix exponential is used to construct a matrix iteration, which propagates the dynamic state of the system step by step on a regular time grid. A large and general class of dynamic inputs to the system, including trains of δ-pulses, can be incorporated into the exact simulation scheme. An extension of the proposed scheme presents an attractive alternative for the approximate simulation of networks of integrate-and-fire neurons with linear sub-threshold integration and non-linear spike generation. The performance of the proposed method is analyzed in comparison with a number of multi-purpose solvers. In simulations of integrate-and-fire neurons, Exact Integration systematically generates the smallest error with respect to both sub-threshold dynamics and spike timing. For the simulation of systems where precise spike timing is important, this results in a practical advantage in particular at moderate integration step sizes. Received: 3 October 1998 / Accepted in revised form: 19 March 1999     

Ladder network prediction of transients in linear spatially inhomogeneous cables

A numerical method is described for finding steady state and transient responses in electrically linear, spatially inhomogeneous cables. Spatial inhomogeneities are incorporated by representing the cable by a number of finite length uniform cylindrical segments, each having the radius and electrical characteristics of a small region along the cable. Input waveforms are approximated by truncated Fourier series of sinusoidal components. Output waveforms are produced by multiplying the input Fourier series sinusoids by their respective transfer functions between input and output points on the cable and summing the resultant output point sinusoids. The transfer functions, representing attenuation and phase shift for each input sinusoid, are obtained by numerical analysis of an electrical ladder network derived from the cylindrical segment model of the cable. Results are shown for application of this method to both cylindrical and expanding radius cable geometries.     

Solution of linear one-dimensional diffusion equations

The paper introduces a basic mathematical form, which is characteristic of a number of linear one-dimensional diffusion equations with coefficients being represented as simple polynomials in the spatial coordinate. A number of particular diffusion equations are introduced and their corresponding exact mathematical solutions are obtained.     

Time-dependent solutions of the spatially implicit neutral model of biodiversity

Previous research into the neutral theory of biodiversity has focused mainly on equilibrium solutions rather than time-dependent solutions. Understanding the time-dependent solutions is essential for applying neutral theory to ecosystems in which time-dependent processes, such as succession and invasion, are driving the dynamics. Time-dependent solutions also facilitate tests against data that are stronger than those based on static equilibrium patterns. Here I investigate the time-dependent solutions of the classic spatially implicit neutral model, in which a small local community is coupled to a much larger metacommunity through immigration. I present explicit general formulas for the eigenvalues, left eigenvectors and right eigenvectors of the models’s transition matrix. The time-dependent solutions can then be expressed in terms of these eigenvalues and eigenvectors. Some of these results are translated directly from existing results for the classic Moran model of population genetics (the Moran model is equivalent to the spatially implicit neutral model after a reparameterization); others of the results are new. I demonstrate that the asymptotic time-dependent solution corresponding to just these first two eigenvectors can be a good approximation to the full time-dependent solution. I also demonstrate the feasibility of a partial eigendecomposition of the transition matrix, which facilitates direct application of the results to a biologically relevant example in which a newly invading species is initially present in the metacommunity but absent from the local community.     

Asymptotic solutions of population dynamic equations

An approach is offered to construct the asymptomatics of the solutions on the small parameter in the close neighborhood of the equilibrium condition of the well-known Volterra-Lotka "prey-predator" system and one of its modifications which takes into account the intraspecies competition of preys and limitation of food resources of a predator. On the basis of the formulas obtained, possible dynamic modes of the size of populations of both kinds are analyzed.     

Changes induced with varying solutions of normal plasma on granulocyte capillary tube migration

In the present work we have evaluated the effect of decreasing concentrations of normal plasma in TC 199 on leukocyte capillary migration. From the 1st to the 4th hour of migration, normal plasma at 100%-80%-60%-40% concentration, significantly inhibits leukocyte migration. The inhibitory effect is lost, only by a few plasmas, at 60% and 40% concentration, during the following migration hours. Normal plasma at 20% and 10% concentration doesn't show any inhibitory effect on leukocyte capillary migration. Therefore, for clinical studies on pathological plasmas we recommend the use o plasma at 20% or 10% concentration to avoid aspecific inhibition.     

Solutions to a degenerate system of parabolic equations from marine biology

Summary A system of parabolic and ordinary differential equations u _t = a ² u _xx + F(u, v, w), v _t = a ² v _xx + G(u, v, w),w _x = – k(u)w is studied which has been proposed by Radach and Maier-Reimer for the dynamics of phytoplankton and nutrient in dependence of light intensity. It is shown that there is a unique solution to this system satisfying given initial and boundary conditions. The solution depends continuously on the data. For specific nonlinearities F, G, and k bounds for the solutions are given.     

Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion

Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv ₁ of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv ₁. Presented at the 1974 SIAM Fall Meeting.