Comment arising from a paper by Wittmer et al.: hypothesis testing for top-down and bottom-up effects in woodland caribou population dynamics

Conservation strategies for populations of woodland caribou Rangifer tarandus caribou frequently emphasize the importance of predator–prey relationships and the availability of lichen-rich late seral forests, yet the importance of summer diet and forage availability to woodland caribou survival is poorly understood. In a recent article, Wittmer et al. (Can J Zool 83:407–418, 2005b) concluded that woodland caribou in British Columbia were declining as a consequence of increased predation that was facilitated by habitat alteration. Their conclusion is consistent with the findings of other authors who have suggested that predation is the most important proximal factor limiting woodland caribou populations (Bergerud and Elliot in Can J Zool 64:1515–1529, 1986; Edmonds in Can J Zool 66:817–826, 1988; Rettie and Messier in Can J Zool 76:251–259, 1998; Hayes et al. in Wildl Monogr 152:1–35, 2003). Wittmer et al. (Can J Zool 83:407–418, 2005b) presented three alternative, contrasting hypotheses for caribou decline that differed in terms of predicted differences in instantaneous rates of increase, pregnancy rates, causes of mortality, and seasonal vulnerability to mortality (Table 1, p 258). These authors rejected the hypotheses that food or an interaction between food and predation was responsible for observed declines in caribou populations; however, the use of pregnancy rate, mortality season and cause of mortality to contrast the alternative hypotheses is problematic. We argue here that the data employed in their study were insufficient to properly evaluate a predation-sensitive foraging hypothesis for caribou decline. Empirical data on seasonal forage availability and quality and plane of nutrition of caribou would be required to test the competing hypotheses. We suggest that methodological limitations in studies of woodland caribou population dynamics prohibit proper evaluation of the mechanism of caribou population declines and fail to elucidate potential interactions between top-down and bottom-up effects on populations. An erratum to this article can be found at     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

A Comparison of Continuous and Discrete-time West Nile Virus Models

The first recorded North American epidemic of West Nile virus was detected in New York state in 1999, and since then the virus has spread and become established in much of North America. Mathematical models for this vector-transmitted disease with cross-infection between mosquitoes and birds have recently been formulated with the aim of predicting disease dynamics and evaluating possible control methods. We consider discrete and continuous time versions of the West Nile virus models proposed by Wonham et al. [Proc. R. Soc. Lond. B 271:501–507, 2004] and by Thomas and Urena [Math. Comput. Modell. 34:771–781, 2001], and evaluate the basic reproduction number as the spectral radius of the next-generation matrix in each case. The assumptions on mosquito-feeding efficiency are crucial for the basic reproduction number calculation. Differing assumptions lead to the conclusion from one model [Wonham, M.J. et al., [Proc. R. Soc. Lond. B] 271:501–507, 2004] that a reduction in bird density would exacerbate the epidemic, while the other model [Thomas, D.M., Urena, B., Math. Comput. Modell. 34:771–781, 2001] predicts the opposite: a reduction in bird density would help control the epidemic.     

Dynamics of annual influenza A epidemics with immuno-selection

 The persistence of Influenza A in the human population relies on continual changes in the viral surface antigens allowing the virus to reinfect the same hosts every few years. The epidemiology of such a drifting virus is modeled by a discrete season-to-season map. During the epidemic season only one strain is present and its transmission dynamics follows a standard epidemic model. After the season, cross-immunity to next year's virus is determined from the proportion of hosts that were infected during the season. A partial analysis of this map shows the existence of oscillations where epidemics occur at regular or irregular intervals. Received: 16 February 2001 / Revised version: 11 June 2002 / Published online: 28 February 2003 Key words or phrases: Infectious disease – Influenza drift – Cross-immunity – Seasonal epidemics – Iterated map     

Estimation of Cell Proliferation Dynamics Using CFSE Data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57–89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1–26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.     

The complete classification for dynamics in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population was proposed by Li et al. in (J Math Biol 10:1037–1052, 1986). The sufficient and necessary conditions for coexistence and the sufficient conditions for stability of the boundary equilibria were provided. This paper will present a thorough classification of dynamics for this model in terms of the first and second so called reproductive numbers of infection in strains I and J. This classification not only solves a conjecture proposed in (Li et al., J Math Biol 10:1037–1052, 1986) but also gives the sufficient and necessary conditions for the competitive exclusion. Supported by the NSF of China grants 10531030 and 10671143.     

Structural principles for periodic orbits in glass networks

A piecewise-linear differential equation model framework for gene regulatory interactions (Glass networks) has allowed considerable analysis of qualitative dynamics in such systems, including periodicity, an important class of regulatory behaviors. Here, we present new results relating the structure of the network to its dynamics (structural principles). The structure we refer to is the state space of the network, which is a digraph on an n-cube in the case of a single threshold per gene. In particular, we show that for a wide class of cycles in the state space there exist parameter values, consistent with the graph structure, for which a periodic orbit exists in the network. For some classes, we show in addition that stable periodic orbits exist. These results extend greatly earlier work by Glass and Pasternack (J Math Biol 6:207–223, 1978).     

A Computational Analysis of Localized Ca<Superscript>2+</Superscript>-Dynamics Generated by Heterogeneous Release Sites

We investigate the role of heterogeneous expression of IP₃R and RyR in generating diverse elementary Ca²⁺ signals. It has been shown empirically (Wojcikiewicz and Luo in Mol. Pharmacol. 53(4):656–662, 1998; Newton et al. in J. Biol. Chem. 269(46):28613–28619, 1994; Smedt et al. in Biochem. J. 322(Pt. 2):575–583, 1997) that tissues express various proportions of IP₃ and RyR isoforms and this expression is dynamically regulated (Parrington et al. in Dev. Biol. 203(2):451–461, 1998; Fissore et al. in Biol. Reprod. 60(1):49–57, 1999; Tovey et al. in J. Cell Sci. 114(Pt. 22):3979–3989, 2001). Although many previous theoretical studies have investigated the dynamics of localized calcium release sites (Swillens et al. in Proc. Natl. Acad. Sci. U.S.A. 96(24):13750–13755, 1999; Shuai and Jung in Proc. Natl. Acad. Sci. U.S.A. 100(2):506–510, 2003a; Shuai and Jung in Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 67(3 Pt. 1):031905, 2003b; Thul and Falcke in Biophys. J. 86(5):2660–2673, 2004; DeRemigio and Smith in Cell Calcium 38(2):73–86, 2005; Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005), so far all such studies focused on release sites consisting of identical channel types. We have extended an existing mathematical model (Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005) to release sites with two (or more) receptor types, each with its distinct channel kinetics. Mathematically, the release site is represented by a transition probability matrix for a collection of nonidentical stochastically gating channels coupled through a shared Ca²⁺ domain. We demonstrate that under certain conditions a previously defined mean-field approximation of the coupling strength does not accurately reproduce the release site dynamics. We develop a novel approximation and establish that its performance in these instances is superior. We use this mathematical framework to study the effect of heterogeneity in the Ca²⁺-regulation of two colocalized channel types on the release site dynamics. We consider release sites consisting of channels with both Ca²⁺-activation and inactivation (“four-state channels”) and channels with Ca²⁺-activation only (“two-state channels”) and show that for the appropriate parameter values, synchronous channel openings within a release site with any proportion of two-state to four-state channels are possible, however, the larger the proportion of two-state channels, the more sensitive the dynamics are to the exact spatial positioning of the channels and the distance between channels. Specifically, the clustering of even a small number of two-state channels interferes with puff/spark termination and increases puff durations or leads to a tonic response.     

Anomalous spreading speeds of cooperative recursion systems

This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.     

On a logical paradox implicit in the notion of a self-reproducing automation

The notion of automaton as used by J. von Neumann is formalized according to methods previously described (Rosen, 1958,Bull. Math. Biophysics 20, 245–60; 317–41). It is observed that a logical paradox arises when one attempts to describe the notion of self-reproducing automaton in this formalism. This paradox is discussed, together with some of the recent attempts to construct automata which exhibit self-reproduction. The relation of these results to biological problems is then investigated.     

Evolutionary tradeoff and equilibrium in an aquatic predator-prey system

Due to the conventional distinction between ecological (rapid) and evolutionary (slow) timescales, ecological and population models have typically ignored the effects of evolution. Yet the potential for rapid evolutionary change has been recently established and may be critical to understanding how populations persist in changing environments. In this paper we examine the relationship between ecological and evolutionary dynamics, focusing on a well-studied experimental aquatic predator-prey system (Fussmann et al., 2000, Science, 290, 1358–1360; Shertzer et al., 2002, J. Anim. Ecol., 71, 802–815; Yoshida et al., 2003, Nature, 424, 303–306). Major properties of predator-prey cycles in this system are determined by ongoing evolutionary dynamics in the prey population. Under some conditions, however, the populations tend to apparently stable steady-state densities. These are the subject of the present paper. We examine a previously developed model for the system, to determine how evolution shapes properties of the equilibria, in particular the number and identity of coexisting prey genotypes. We then apply these results to explore how evolutionary dynamics can shape the responses of the system to ‘management’: externally imposed alterations in conditions. Specifically, we compare the behavior of the system including evolutionary dynamics, with predictions that would be made if the potential for rapid evolutionary change is neglected. Finally, we posit some simple experiments to verify our prediction that evolution can have significant qualitative effects on observed population-level responses to changing conditions.     

The Repair Response to Osteochondral Implant Types in a Rabbit Model

Current treatments for damaged articular cartilage (i.e., shaving the articular surface, perforation or abrasion of the subchondral bone, and resurfacing with periosteal and perichondrial resurfacing) often produce fibrocartilage, or hyaline-appearing repair that is not sustained over time (Henche 1967, Ligament and Articular Cartilage Injuries. Springer-Verlag, New York, NY, pp. 157–164; Insall 1974, Clin. Orthop. 101: 61–67; Mitchell and Shepard 1976, J. Bone Joint Surg. [Am.] 58: 230–233; O’Driscoll et al. 1986, J. Bone Joint Surg. [Am.] 68: 1017–1035; 1989, Trans. Orthop. Res. Soc. 14: 145; Kim et al. 1991, J. Bone Joint Surg. [Am.] 73: 1301–1315). Autologous chondrocyte transplantation, although promising, requires two surgeries, has site-dependent and patient age limitations, and has unknown long-term donor site morbidity (Brittberg et al. 1994, N Engl. J. Med. 331: 889–895; Minas 2003, Orthopedics 26: 945–947; Peterson et al. 2003, J. Bone Joint Surg. Am. 85-A(Suppl. 2): S17–S24). Osteochondral allografts remain a widely used method of articular resurfacing to delay arthritic progression. The present study compared the histological response to four types of osteochondral implants in a rabbit model: autograft, frozen, freeze-dried, and fresh implants. Specimens implanted in the femoral groove were harvested at 6 and 12 weeks. Results showed similar restoration of the joint surface regardless of implant type, with a trend toward better repair at the later timepoint. As has been observed in other studies (Frenkel et al. 1997, J. Bone Joint Surg. 79B: 281–286; Toolan et al. 1998, J. Biomed. Mater. Res. 41: 244–250), each group in this study had at least one specimen in which a healthy-appearing surface on the implant was not well-integrated with host tissues. Although the differences were not statistically significant, freeze-dried implants at both timepoints had the best histological scores. The osteochondral grafts tested successfully restored the gross joint surface and congruity. At 12 weeks, no significant differences were observed between the various allografts and autologous osteochondral grafts.     

Phytoplankton blooms and fish recruitment rate: Effects of spatial distribution

We consider the spatio-temporal dynamics of a spatially-structured generalization of the phytoplankton-zooplankton-fish larvae model system proposed earlier (Biktashev et al., 2003, J. Plankton Res. 5, 21–33; James et al., 2003, Ecol. Model. 160, 77–90). In contrast to Pitchford and Brindley (2001, Bull. Math. Biol. 63, 527–546), who were concerned with small scale patchiness (i.e., 1–10m), on which the (stochastic) raptorial behaviour of individual larvae is important, we address here the much larger scale ‘patchy’ problems (i.e., 10–100 km), on which both larvae and plankton may be regarded as passive tracers of the fluid motion, dispersed and mixed by the turbulent diffusion processes. In particular, we study the dependence of the fish recruitment on carrying capacities of the plankton subsystem and on spatio-temporal evolution of that subsystem with respect to the larvae hatching site(s). It is shown that the main features found both in the nonstructured and age-structured spatially uniform models are observed in the spatially structured case, but that spatial effects can significantly modify the overall quantitative outcome. Spatial patterns in the metamorphosed fish distribution are a consequence of quasi-local interaction of larvae with plankton, in which the dispersion of larvae by large scale turbulent eddies plays little part due to the relatively short timescale of the larvae development. As a result, in a strong phyto/zooplankton subsystem, with fast reproduction rate and large carrying capacity of phytoplankton and high conversion ratio of zooplankton, recruitment success depends only on the localization and timing of the hatching with respect to the plankton patches. In a weak phyto/zooplankton system, with slow reproduction rate and small carrying capacity of phytoplankton and low conversion ratio of zooplankton, the larvae may significantly influence the evolution of the plankton patches, which may lead to nontrivial cooperative effects between different patches of larvae. However, in this case, recruitment is very low.     

Explaining how and explaining why: developmental and evolutionary explanations of dominance

There have been two different schools of thought on the evolution of dominance. On the one hand, followers of Wright [Wright S. 1929. Am. Nat. 63: 274–279, Evolution: Selected Papers by Sewall Wright, University of Chicago Press, Chicago; 1934. Am. Nat. 68: 25–53, Evolution: Selected Papers by Sewall Wright, University of Chicago Press, Chicago; Haldane J.B.S. 1930. Am. Nat. 64: 87–90; 1939. J. Genet. 37: 365–374; Kacser H. and Burns J.A. 1981. Genetics 97: 639–666] have defended the view that dominance is a product of non-linearities in gene expression. On the other hand, followers of Fisher [Fisher R.A. 1928a. Am. Nat. 62: 15–126; 1928b. Am. Nat. 62: 571–574; Bürger R. 1983a. Math. Biosci. 67: 125–143; 1983b. J. Math. Biol. 16: 269–280; Wagner G. and Burger R. 1985. J. Theor. Biol. 113: 475–500; Mayo O. and Reinhard B. 1997. Biol. Rev. 72: 97–110] have argued that dominance evolved via selection on modifier genes. Some have called these “physiological” versus “selectionist,” or more recently [Falk R. 2001. Biol. Philos. 16: 285–323], “functional,” versus “structural” explanations of dominance. This paper argues, however, that one need not treat these explanations as exclusive. While one can disagree about the most likely evolutionary explanation of dominance, as Wright and Fisher did, offering a “physiological” or developmental explanation of dominance does not render dominance “epiphenomenal,” nor show that evolutionary considerations are irrelevant to the maintenance of dominance, as some [Kacser H. and Burns J.A. 1981. Genetics 97: 639–666] have argued. Recent work [Gilchrist M.A. and Nijhout H.F. 2001. Genetics 159: 423–432] illustrates how biological explanation is a multi-level task, requiring both a “top-down” approach to understanding how a pattern of inheritance or trait might be maintained in populations, as well as “bottom-up” modeling of the dynamics of gene expression.     

An interstitial telomere array proximal to the distal telomere of mouse chromosome 13

Lambda clones of mouse DNA from BALB/c and C57BL/10, each containing an array of telomere hexamers, were localized by FISH to a region close to the telomere of Chr 13. Amplification of mouse genomic DNA with primers flanking SSRs within the cloned DNA showed several alleles, which were used to type eight sets of RI strains. The two lambda clones contained allelic versions of the interstitial telomere array, Tel-rs4, which is 495 bp in C57BL/10 and which includes a variety of sequence changes from the consensus telomere hexamer. Comparison of the segregation of the amplification products of the SSRs with the segregation of other loci in an interspecies backcross (C57BL/6JEi × SPRET/Ei) F₁× SPRET/Ei shows recombination suppression, possibly associated with ribosomal DNA sequences present on distal Chr 13 in Mus spretus, when compared with recombination in an interstrain backcross, (C57BL/6J × DBA/J) F₁× C57BL/6J, and with the MIT F₂ intercross. Analysis of recombination in females using a second interstrain backcross, (ICR/Ha × C57BL/6Ha) F₁× C57BL/6Ha, also indicates recombination suppression when compared with recombination in males of the same strains, using backcross C57BL/6Ha × (ICR/Ha × C57BL/6Ha) F₁. Thus, more than one cause may contribute to recombination suppression in this region. The combined order of the loci typed was D13Mit37–D13Mit30–D13Mit148–(D13Rp1, 2, 3, 4, Tel-rs4)–D13Mit53–D13Mit196–D13Mit77–(D13Mit78, 35). Data from crosses where apparently normal frequencies of recombination occur suggest that the telomere array is about 6 map units proximal to the most distal loci on Chr 13. This distance is consistent with evidence from markers identified in two YAC clones obtained from the region. Received: 24 September 1996/Accepted: 20 January 1997     

Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form. Received: 26 January 1999 / Revised version: 20 October 2000 / Published online: 10 April 2001