Periodic orbits near heteroclinic cycles in a cyclic replicator system

A species is semelparous if every individual reproduces only once in its life and dies immediately after the reproduction. While the reproduction opportunity is unique per year and the individual’s period from birth to reproduction is just n years, the individuals that reproduce in the ith year (modulo n) are called the ith year class, i = 1, 2, . . . , n. The dynamics of the n year-class system can be described by a differential equation system of Lotka–Volterra type. For the case n = 4, there is a heteroclinic cycle on the boundary as shown in previous works. In this paper, we focus on the case n = 4 and show the existence, growth and disappearance of periodic orbits near the heteroclinic cycle, which is a part of the conjecture by Diekmann and van Gils (SIAM J Appl Dyn Syst 8:1160–1189, 2009). By analyzing the Poincaré map near the heteroclinic cycle and introducing a metric to measure the size of the periodic orbit, we show that (i) when the average competitive degree among subpopulations (year classes) in the system is weak, there exists an asymptotically stable periodic orbit near the heteroclinic cycle which is repelling; (ii) the periodic orbit grows in size when some competitive degree increases, and converges to the heteroclinic cycle when the average competitive degree tends to be strong; (iii) when the average competitive degree is strong, there is no periodic orbit near the heteroclinic cycle which becomes asymptotically stable. Our results provide explanations why periodic solutions expand and disappear and why all but one subpopulation go extinct.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Species abundance distributions in neutral models with immigration or mutation and general lifetimes

We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate λ. Thus, the population size is a homogeneous, binary Crump–Mode–Jagers process (which is not necessarily a Markov process). We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate μ (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at constant rate θ. We are interested in the species abundance distribution, i.e., in the numbers, denoted I _n (k) in the immigration model and A _n (k) in the mutation model, of species represented by k individuals, k = 1, 2, . . . , n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I _t (k); k ≥ 1) of species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher’s log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by Ewens’ sampling formula. In particular, I _n (k) converges as n → ∞ to a Poisson r.v. with mean γ/k, where γ : = μ/λ. In the mutation model, as n → ∞, we obtain the almost sure convergence of n ⁻¹ A _n (k) to a nonrandom explicit constant. In the case of a critical, linear birth–death process, this constant is given by Fisher’s log-series, namely n ⁻¹ A _n (k) converges to α ^k /k, where α : = λ/(λ + θ). In both models, the abundances of the most abundant species are briefly discussed.     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.     

Existence of traveling wave solutions in a diffusive predator-prey model

 We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R ⁴ and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R ⁴ . The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem. Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002 RID="*" ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China. RID="*" ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. Mathematics Subject Classification (2000): 34C35, 35K57 Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

The impact of year-to-year changes in the weather on the dynamics of Daphnia in a thermally stratified lake

The factors influencing the seasonal dynamics of Daphnia in a thermally stratified lake (Esthwaite Water) are described and related to long-term changes in the weather. The Daphnia produced three cohorts in the year and the strength of the cohorts was determined by year-to-year variations in the physical characteristics of the lake and the abundance of edible algae. Food was most abundant in early summer when small, fast-growing flagellates were particularly common. In late summer, the phytoplankton community was dominated by large, inedible species but edible forms re-appeared when nutrients were entrained by wind mixing. Examples are presented to demonstrate the effect that year-to-year variations in the weather have on the growth of the phytoplankton and the dynamics of the Daphnia. In ‘good’ years, when the lake stratifies early and there are periods of episodic mixing in summer, there are two ‘pulses’ of edible algae and two strong cohorts of Daphnia. In ‘bad’ years when stratification is delayed and there is little episodic mixing, the growth of the edible algae is suppressed and the Daphnia produce two weak cohorts. The results are discussed in relation to the impact of intermediate disturbances on growth of phytoplankton and current theories of population regulation in Daphnia. The evidence suggests that the dynamics of the Daphnia in the lake are strongly influenced by seasonal variations in the mixing regime, the recycling of nutrients and the episodic growth of edible algae.     

Mathematical models for cellular systems. The von foerster equation. Part II

This is the continuation of Part I, which was published in the September, 1965, issue of theBulletin. The birth rate, α(t), is now assumed to be a linear functional of the age density,n. This gives a simple model of self-replenishing stem cell compartments, and leads to a necessary condition for the existence of a steady state. Some examples are presented to illustrate the formalism. They include: (a) An equivivant population with life spanD and no losses from death or migration. The total number of cells is multiplied by 2 in each time intervalD. As a special case, frequently realized in practice, the population may be increasing exponentially with time (“log-phase” of growth). (b) A compartment with “random” emigration of cells and gamma distribution of life spans. (c) An oversimplified version of L. G. Lajtha’s model describing stem cell kinetics. In section IV a simple case in which the loss function depends explicitly onn is discussed very briefly. This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

Enhancement of biodesulfurization by <Emphasis Type="Italic">Pseudomonas delafieldii</Emphasis> in a ceramic microsparging aeration system

A 700 ml membrane-aerated, stirred glass reactor equipped with four vertical baffles was constructed. Biodesulfurization of model oil (n-dodecane containing dibenzothiophene—DBT) and hydrodesulfurized diesel was carried out using Pseudomonas delafieldii strain R-8. Microbubble aeration gave an activity of 1.3 mg DBT removed g⁻¹ h⁻¹ and 277 μg sulfur g⁻¹ h⁻¹ for model oil and hydrodesulfurized diesel, respectively. These values were 1.9- and 1.6-times higher than using a traditional bubble aeration process. This is a promising method for the biodesulfurization of petroleum feedstocks.     

Force–velocity relationship for multiple kinesin motors pulling a magnetic bead

Although the velocity of single kinesin motors against an opposing force F of 0–10 pN is well known, the behavior of multiple kinesin motors working to overcome a larger load is still poorly understood. We have carried out gliding assays in which 3–7 Drosophila kinesin-1 motors moved a microtubule at 200–700 μm/s against a 0–31 pN load at saturating [ATP]. The load F was generated by applying a spatially uniform magnetic field gradient to a superparamagnetic bead attached to the (+) end of the microtubule. When F was scaled by the average number of motors 〈n〉, the force–velocity relationship for multiple motors was similar to the force–velocity relationship for a single motor, supporting a minimal load-sharing model. The velocity distribution at low load has a single mode consistent with rapid fluctuations of n. However, against a load of 2.5–4.7 pN/motor, additional modes appeared at lower velocity. These observations support the Klumpp–Lipowsky model of multimotor transport [Proc Natl Acad Sci USA 102. 17284–17289 (2005)].     

Negative effects of habitat drying and prior exploitation on the detritus resource in an ephemeral aquatic habitat

Ephemeral aquatic habitats are characterized by cycles of drying and subsequent inundation, and by production of sequential non-overlapping cohorts of organisms. Both processes may alter the quantity or quality of resources, and may therefore affect survival and development of cohorts that subsequently colonize ephemeral habitats. We examined these effects of habitat drying and non-overlapping cohorts on experimental cohorts of the tree hole mosquito Aedes triseriatus, testing specifically whether the value of leaf litter as a food resource is altered by cycles of inundation and drying, or by exploitation by a prior non-overlapping cohort. We created four treatments of leaf litter: (1) no prior cohort, continuously wet; (2) no prior cohort, one␣wet/dry cycle; (3) prior cohort, continuously wet, and (4) prior cohort, one wet/dry cycle, and tested for effects on individual fitness components (survivorship, mean dry mass at, and median days to eclosion) and on population growth (estimated finite rate of increase –λ′). Both resource drying and the presence of a prior cohort negatively affected individual fitness components in tires, increasing days to eclosion, and decreasing mean dry mass at eclosion for both sexes. Resource drying also negatively affected estimated rates of increase (λ′) in tree holes. A prior cohort had no significant effects on λ′. These results indicate that intraspecific interactions among mosquito larvae may include amensalistic effects of earlier, non-overlapping cohorts, and that resource drying reduces resource quality. The latter effect indicates that enhanced production of A. triseriatus from recently filled containers is not due to resource drying per se, and may result from more complex community-level effects of habitat drying. Extreme cycles of drying and inundation seem likely to increase intraspecific resource competition among drought-adapted species like A. triseriatus. Received: 5 May 1997 / Accepted: 20 January 1998     

A mathematical model for drug administration by using the phagocytosis of red blood cells

 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. Received 15 December 1994; received in revised form 20 July 1995     

Reduction of biomass in a bioscrubber for waste gas treatment by limited supply of phosphate and potassium ions

  Elimination of n-butanol from the gas phase was examined with a mixed culture in a compact bioscrubber. The extent of the cell concentration was limited by the supply of n-butanol, phosphate or potassium, and the growth rate was determined by the dilution rate. With n-butanol as the limiting substrate the cellular yield was 0.53 g dry cell weight/g n-butanol. Phosphate limitation decreased this yield to 0.34 g and potassium limitation to 0.31 g dry cell weight/g n-butanol at a dilution rate of 0.1/h. Under these conditions n-butanol was eliminated from the gas phase by 84%–100%. In the same order of limitations the specific degradation rate ranged from 0.19 g to 0.32 g n-butanol g dry cell weight⁻¹ h⁻¹. The fraction of n-butanol required to satisfy the needs for maintenance energy increased significantly depending on the limiting nutrient. Limitation by n-butanol, phosphate or potassium caused a maintenance requirement of 0.07, 0.16 and 0.34 g n-butanol g dry cell weight⁻¹ h⁻¹, thus showing a fivefold increase. This high demand for the carbon source demonstrated the feasibility of operating a bioscrubber under mineral limitation to reduce biomass formation significantly, and to maintain a high degree of substrate elimination from the gas phase. Received: 22 May 1996 / Received revision: 23 July 1996 / Accepted: 5 August 1996     

Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

An immortal cell line to study the role of endogenous cftr in electrolyte absorption

Summary The intact human reabsorptive sweat duct (RD) has been a reliable model for investigations of the functional role of “endogenous” CFTR (cystic fibrosis transmembrane conductance regulator) in normal and abnormal electrolyte absorptive function. But to overcome the limitations imposed by the use of fresh, intact tissue, we transformed cultured RD cells using the chimeric virus Ad5/SV40 1613 ori-. The resultant cell line, RD2(NL), has remained differentiated forming a polarized epithelium that expressed two fundamental components of absorption, a cAMP activated Cl⁻ conductance (G_cl) and an amiloride-sensitive Na⁺ conductance (G_Na). In the unstimulated state, there was a low level of transport activity; however, addition of forskolin (10⁻⁵ M) significantly increased the Cl⁻ diffusion potential (V_t) generated by a luminally directed Cl⁻ gradient from − 15.3 ± 0.7 mV to −23.9 ± 1.1 mV,n=39; and decreased the transepithelial resistance (R_t) from 814.8 ± 56.3 Ω.cm² to 750.5 ± 47.5 Ω.cm²,n=39, (n=number of cultures). cAMP activation, anion selectivity (Cl⁻>I⁻>gluconate), and a dependence upon metabolic energy (metabolic poisoning inhibited G_Cl), all indicate that the G_Cl expressed in RD2(NL) is in fact CFTR-G_Cl. The presence of an apical amiloride-sensitive G_Na was shown by the amiloride (10⁻⁵ M) inhibition of G_Na as indicated by a reduction of V_t and equivalent short circuit current by 78.0 ± 3.1% and 77.9 ± 2.6%, respectively, and an increase in R_t by 7.2 ± 0.8%,n=36. In conclusion, the RD2(NL) cell line presents the first model system in which CFTR-G_Cl is expressed in a purely absorptive tissue. It provides an opportunity to study the properties and role of CFTR in the context of absorptive function in immortalized epithelial cells.     

Survival Chances of Mutants Starting With One Individual

A simple theoretical model of a Darwinian system (a periodic system with a multiplication phase and a selection phase) of entities (initial form of polymer strand, primary mutant and satellite mutants) is given. First case: one mutant is considered. One individual of the mutant appears in the multiplication phase of the first generation. The probabilities to find N mutants W_n^M(N) after the multiplication phase M of the n-th generation (with probability δ of an error in the replication, where all possible errors are fatal errors) and W_n^S(N) after the following selection phase S (with probability β that one individual survives) are given iteratively. The evolutionary tree is evaluated. Averages from the distributions and the probability of extinction W_∞^S(0) are obtained. Second case: two mutants are considered (primary mutant and new form). One individual of the primary mutant appears in the multiplication phase of the first generation. The probabilities to find N_p primary mutants and N_m of the new form W_n^M(N_p, N_m) after the multiplication phase M of the n-th generation (probability ε of an error in the replication of the primary mutant giving the new form) and W_n^S(N_p, N_m) after the following selection phase S (probabilities β_p and β_m that one individual each of the primary mutant and of the new form survives) are given iteratively. Again the evolutionary tree is evaluated. Averages from the distributions are obtained.     

An Emulsion of Sulfoquinovosylacylglycerol with Long-Chain Alkanes Increases Its Permeability to Tumor Cells

The α-anomer form of sulfoquinovosyl-monoacylglycerol with a saturated C18 fatty acid (α-SQMG-C_18:0) is a natural sulfolipid that is a clinically promising antitumor agent. It forms vesicles, micelles or an emulsion in water, depending on several physicochemical conditions. The type of aggregate formed appears to strongly influence the bioactivity level. Thus, we investigated the nature of the aggregates in relation to their bioactivities. The structure of the α-SQMG-C_18:0 assembly was greatly affected by the type of additive used in the preparation. Emulsification with ethanol and n-decane might be more effective at inhibiting tumor cell growth than the micelle or vesicle preparations. α-SQMG-C_18:0 formed an “emulsion-like-aggregate” in ethanol containing an n-decane concentration in the range of 1.03–103 mM. These ethanol/n-alkane/α-SQMG-C_18:0 aggregates inhibited cell growth in a dose-dependent manner, under optimum conditions (i.e., ethanol containing 103 mM of n-decane or n-dodecane dispersed in phosphate-buffered saline or culture medium). Based on these data, we discuss the relationship between the molecular action of and antitumor activity by α-SQMG-C_18:0.