Advection–diffusion equations for generalized tactic searching behaviors

 Many organisms search for limiting resources by using repeated responses to local cues, which cumulatively cause movement towards more favorable parts of their environment. This paper presents a general asymptotic expression, derived under the assumption of shallow environmental gradients, for the population-level flux of organisms moving at a constant speed and reorienting at rates determined by the environmental conditions experienced since the last reorientation. The expression takes the form of an advection-diffusion equation, in which the diffusivity and advection velocity are determined by statistics of the turning algorithm that are directly comparable to empirical observations. This work provides a mechanism with which to systematically evaluate a wide variety of tactic and kinetic strategies for determining turning behaviors. The model is applied to searchers on spatially-variable, random distributions of discrete resource patches. Such algorithms are functions of the integrated resource density encountered between turns. It is shown that behaviors in which the turning time distribution is a function of integrated density cannot result in taxis. In contrast, behaviors in which the turning rate is a function of integrated density can result in taxis. These two classes of search algorithm differ in that the latter requires the searcher to “learn” about its local environment, whereas the former requires no such assessment. This suggests neural or physiological mechanisms for remembering previous encounters may be a biological requirement for searchers on discrete resource distributions. Received: 21 September 1995/Revised version: 18 July 1996     

Are basal metabolic rate prediction equations appropriate for female children and adolescents?

Wong, William W., Nancy F. Butte, Albert C. Hergenroeder,Rebecca B. Hill, Janice E. Stuff, and E. O'Brian Smith. Are basalmetabolic rate prediction equations appropriate for female children andadolescents? J. Appl. Physiol. 81(6):2407-2414, 1996.The basal metabolic rate (BMR), which accountsfor 50-70% of total energy expenditure, is essential forestimation of patient and population energy needs. Numerous equationshave been formulated for prediction of human BMR. Most equations incurrent use are based on measurements of Caucasians performed more thanfour decades ago. We evaluated 10 prediction equations commonly usedfor estimation of BMR in 76 Caucasian and 42 African-American girlsbetween 8 and 17 yr of age against BMR measured by whole-bodycalorimetry. The majority of the prediction equations (9 of10) overestimated BMR by 60 ± 46 kcal/day (range,15-176 kcal/day). This overestimation was found to besignificantly greater (P < 0.05) forAfrican-Americans (77 ± 17 kcal/day) than for Caucasians (25 ± 17 kcal/day) in six equations, controlling for age, weight, and sexualmaturity. We conclude that ethnicity is an important factor inestimation of the BMR and that the current prediction equations are notappropriate for accurate estimation of the BMR of individual femalechildren and adolescents.

     

Qualitative permanence of Lotka–Volterra equations

In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.     

“Exchange diffusion”: Rate equations for the influx of α-aminoisobutyric acid into mouse cerebrum slices containing this amino acid

Rate equations for the gross influx of -aminoisobutyric acid (AIB) into mouse cerebrum slices containing AIB have a first-order term for unsaturable concentrative influx, identical to the corresponding term for unloaded slices, and a modified Michaelis-Menten term,V_max/(1+K _t /S), for saturable concentrative influx. [V_max v _L (1+K _t /S), wherev _L =saturable component of influx,S=AIB concentration in medium, andK _t =Michaelis constant for unloaded slices.] Below a tissue AIB (T) of 19 µmol/g final wet weight,V_max increases linearly followingV_max=V ₁+m ₁ T; above that value,V _max is virtually constant. The transition is sharp. This equation is consistent with a carrier model for active transport. At the transition, intracellular AIB is about 1 molecule for every 70 amino acid residues of tissue protein, vastly more than could be accommodated by AIB-binding sites in cell membranes. The transition may come from a slow process that does not fill all sites when the tissue AIB is below the transition concentration, or from an AIB-induced phase transition in the membrane.Nomenclature AIB -aminoisobutyric acid - A radioactivity of reference; unspecified amino acid - C counts in tissue sample; carrier for transport - C _i carrier in form that reacts with intracellular substrate - C _o carrier in form that reacts with extracellular substrate - C _R counts in reference - CS complex of substrate with carrier - (CS) _i complex of substrate with carrier in formC _i - (CS) _o complex of substrate with carrier in formC _o - G counts per gram of tissue - HEPES N-2-Hydroxyethylpiperazine-N-2-ethanesulfonic acid - k _u rate constant for first-order unsaturable uptake - K,K ,K ,K ,K _d adjustable parameters in Eqs. (9)–(13) for v, analogous to the Michaelis constant - K _d dissociation constant - K _t Michaelis constant for saturable uptake - K _t Michaelis constant for gross saturable uptake by tissue containing substrate - m ₁,m ₂ slope in Eq. (5) or (6) expressing dependence ofV_max onT orT _i ^w in Region 1 or 2 - M binding site for amino acid A - n number of data points - P number of parameters to be determined; parameter in Stein's (1981) equation, Eq. (17) in this paper - P ₁,P ₂,P ₁₂ property of tissue with unoccupied binding sites, property of tissue with occupied binding sites, property of tissue with both unoccupied and occupied binding sites, respectively - Q parameter in Stein's (1981) equation, Eq. (17) in this paper - r Pearson's correlation coefficient - Relative error RE =100{[(observed quantity – calculated quantity)/calculated quantity]²/(n –P)}^1/2 - S concentration of substrate in medium; transport substrate - S _i intracellular transport substrate - S _int AIB in medium corresponding to intracellular AIB at intersection - S _o extracellular transport substrate - T observed concentration of substrate in tissue including substrate in extracellular space and adherent fluids - T _i intracellular concentration of substrate - T _int tissue AIB corresponding to intracellular AIB at intersection - T _i ^w ,T _i ^/30 intracellular concentration of substrate withw% (30%) extracellular and adherent fluids - U observed uptake of labeled substrate by incubated tissue including substrate in extracellular and adherent fluids - U _R observed uptake of labeled substrate referred to concentration of substrate in medium - U _max adjustable parameter in Eqs. (9)–(15) for v, analogous to the Michaelis-Menten maximum rate,V _max - v influx of substrate - v _L gross influx of substrate into tissue containing substrate - v _L contribution of saturable component to gross influx into tissue containing substrate - v incremental influx, that is, gross influx into tissue that contains substrate minus influx under the same conditions into tissue that does not contain substrate - V ₁,V ₂ intercept in Eq. (5) or (6) expressing dependence ofV_max onT orT _i ^w in Region 1 or 2, respectively - V _max maximum rate in Michaelis-Menten equation - V_max apparent maximum rate defined byV_maxv_max(1+K _t /S) - V_{max 1},V_{max 2} apparent maximum rate in Region 1 or 2, respectively - V_int apparent maximum rate at intersection defining boundary between Regions 1 and 2 - w weight of incubated tissue - W _d dry weight of tissue expressed as fraction by weight - W _e extracellular and surface space of incubated tissue expressed as percent by weight - , , adjustable parameters in modified expressions for gross unsaturable influx into tissue containing substrate - , , , exponents ofS orT in Eqs. (9)–(13) for v - parameter in Stein's (1981) equation, Eq. (17), corresponding more or less tom ₁ For my wife, Lynn.     

Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles

The paper presents a qualitative analysis of the following systems ofn differential equations: \(\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )\) , which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.     

Sensitivity analysis of the Poisson Nernst–Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst–Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

     

Rotating wave solutions of the FitzHugh–Nagumo equations

This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.     

Disparity estimation through Green’s functions of matching equations

Binocular disparities arise from positional differences of scene features projected in the two retinae, and constitute the primary sensory cue for stereo vision. Here we introduce a new computational model for disparity estimation, based on the Green’s function of an image matching equation. When filtering a Gabor-function-modulated signal, the considered Green’s function yields a similarly modulated but shifted version of the original signal. Since a Gabor function models the receptive field of a cortical simple cell, the Green’s kernel thus allows the simulation of relative shifts between the cell’s left and right binocular inputs. A measure of the local degree of matching of such shifted inputs can then be introduced which affords disparity estimation in a similar manner to the energy model of the complex cortical cells. We have therefore effectively reformulated, in physiologically plausible terms, an image matching approach to disparity estimation. Our experiments show that the Green’s function method allows the detection of disparities both from random-dot and real-world stereograms. Partially supported by CNPq-Brazil.     

A nonlinear autoregressive Volterra model of the Hodgkin–Huxley equations

We propose a new variant of Volterra-type model with a nonlinear auto-regressive (NAR) component that is a suitable framework for describing the process of AP generation by the neuron membrane potential, and we apply it to input-output data generated by the Hodgkin–Huxley (H–H) equations. Volterra models use a functional series expansion to describe the input-output relation for most nonlinear dynamic systems, and are applicable to a wide range of physiologic systems. It is difficult, however, to apply the Volterra methodology to the H–H model because is characterized by distinct subthreshold and suprathreshold dynamics. When threshold is crossed, an autonomous action potential (AP) is generated, the output becomes temporarily decoupled from the input, and the standard Volterra model fails. Therefore, in our framework, whenever membrane potential exceeds some threshold, it is taken as a second input to a dual-input Volterra model. This model correctly predicts membrane voltage deflection both within the subthreshold region and during APs. Moreover, the model naturally generates a post-AP afterpotential and refractory period. It is known that the H–H model converges to a limit cycle in response to a constant current injection. This behavior is correctly predicted by the proposed model, while the standard Volterra model is incapable of generating such limit cycle behavior. The inclusion of cross-kernels, which describe the nonlinear interactions between the exogenous and autoregressive inputs, is found to be absolutely necessary. The proposed model is general, non-parametric, and data-derived.     

Non-periodicity in chemostat equations: a multi-dimensional negative Bendixson–Dulac criterion

We establish conditions which exclude periodic solutions in a simple chemostat with a single nutrient and N competing species. Growth rates are not required to be proportional to food uptake. Instead of a Lyapunov function approach, we develop and apply a multi-dimensional Bendixson–Dulac type exclusion principle based on differential forms.      

A parameter uniform essentially firsorder convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction—diffusion type with initial and Robin boundary conditions

In this paper,a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered.The components of the solution u→ of this system are smooth,whereas the components of αu→/αx exhibit parabolic boundary layers.A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested.This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.     

Information for Authors

Acta Zoologica Sinica is a bimonthly, peer-reviewed journal and publishes original articles, reviews, viewpoints, research methods, information about academic conferences and reviews on new books in all aspects of Zoology, and particularly welcome contributions in research fields of ecology,behaviorial biology, evolutionary biology and biogeography.     

Information for Authors

Acta Zoologica Sinica is a bimonthly, peer-reviewed journal and publishes original articles, reviews, viewpoints, research methods, information about academic conferences and reviews on new books in all aspects of Zoology, and particularly welcome contributions in research fields of ecology, behaviorial biology, evolutionary biology and biogeography.     

Information for Authors

Acta Zoologica Sinica is a bimonthly, peer-reviewed journal and publishes original articles, reviews, viewpoints, research methods, information about academic conferences and reviews on new book sin all aspects of Zoology, and particularly welcome contributions in research fields of ecology, behaviorial biology, evolutionary biology and biogeography.     

Information for authors

Science in China Series C: Life Sciences, an academic journal cosponsored by the Chinese Academy of Sciences and the National Natural Science Foundation of China, and p