Theor Biol Research Articles

A Mathematical Model of Fluid Transport in an Accurate Reconstruction of Parotid Acinar Cells.

Salivary gland acinar cells use the calcium ([Formula: see text]) ion as a signalling messenger to regulate a diverse range of intracellular processes, including the secretion of primary saliva. Although the underlying mechanisms responsible for saliva secretion are reasonably well understood, the precise role played by spatially heterogeneous intracellular [Formula: see text] signalling in these cells remains uncertain. In this study, we use a mathematical model, based on new and unpublished experimental data from parotid acinar cells (measured in excised lobules of mouse parotid gland), to investigate how the structure of the cell and the spatio-temporal properties of [Formula: see text] signalling influence the production of primary saliva. We combine a new [Formula: see text] signalling model [described in detail in a companion paper: Pages et al. in Bull Math Biol 2018, submitted] with an existing secretion model (Vera-Sigüenza et al. in Bull Math Biol 80:255-282, 2018. https://doi.org/10.1007/s11538-017-0370-6 ) and solve the resultant model in an anatomically accurate three-dimensional cell. Our study yields three principal results. Firstly, we show that spatial heterogeneities of [Formula: see text] concentration in either the apical or basal regions of the cell have no significant effect on the rate of primary saliva secretion. Secondly, in agreement with previous work (Palk et al., in J Theor Biol 305:45-53, 2012. https://doi.org/10.1016/j.jtbi.2012.04.009 ) we show that the frequency of [Formula: see text] oscillation has no significant effect on the rate of primary saliva secretion, which is determined almost entirely by the mean (over time) of the apical and basal [Formula: see text]. Thirdly, it is possible to model the rate of primary saliva secretion as a quasi-steady-state function of the cytosolic [Formula: see text] averaged over the entire cell when modelling the flow rate is the only interest, thus ignoring all the dynamic complexity not only of the fluid secretion mechanism but also of the intracellular heterogeneity of [Formula: see text]. Taken together, our results demonstrate that an accurate multiscale model of primary saliva secretion from a single acinar cell can be constructed by ignoring the vast majority of the spatial and temporal complexity of the underlying mechanisms.

Analysis of Age-Structured Pertussis Models with Multiple Infections During a Lifetime

One of the unique features of bacterial diseases such as pertussis is the possibility of multiple infections during a lifetime. Immunity gained after each infection may play an important role in disease transmission dynamics. A PDE model with two infections for pertussis was considered in Feng et al. (J Theor Biol 356:123–132, 2014) and applied to the Swedish population to estimate the age-dependent probability of infection on contact. However, no detailed analysis of the dynamic properties of the model was considered in that study. Here we present the analysis including existence and stability of equilibrium solutions, which are shown to be determined by the effective (or basic) reproduction number $${\mathcal {R}}$$ (or $${\mathcal {R}}_0$$). We also extend the model in Feng et al. (2014) to allow three infections during a lifetime. And we derive the age-specific probability of infection during a lifetime, denoted by F(a), using two approaches: one uses the model solutions, and the other one is based on biological interpretations. Numerical explorations of the model suggest that, with reasonable assumptions, two infections are all that one needs to consider to account for the dynamics of this disease. This increases the importance of the analytic results from the 2-infection model. Nonetheless, the formula for F(a) in the 3-infection model may provide more accurate description for the probability of infection because it permits more realistic distributions for waiting times in disease stages.

Non-triangular cross-diffusion systems with predator–prey reaction terms

A predator–prey system involving cross-diffusion is obtained at the formal level as a singular limit of a four-species reaction–diffusion system, following the approach proposed in the context of ODEs by Geritz and Gyllenberg (J Theor Biol 314:106–108, 2012). Part of this derivation can be made rigorous. The possibility of appearance of Turing patterns for this cross-diffusion system is studied, and compared to what happens when standard diffusion terms replace the cross-diffusion terms.

An Alternative Formulation for a Distributed Delayed Logistic Equation

We study an alternative single species logistic distributed delay differential equation (DDE) with decay-consistent delay in growth. Population oscillation is rarely observed in nature, in contrast to the outcomes of the classical logistic DDE. In the alternative discrete delay model proposed by Arino et al. (J Theor Biol 241(1):109-119, 2006), oscillatory behavior is excluded. This study adapts their idea of the decay-consistent delay and generalizes their model. We establish a threshold for survival and extinction: In the former case, it is confirmed using Lyapunov functionals that the population approaches the delay modified carrying capacity; in the later case the extinction is proved by the fluctuation lemma. We further use adaptive dynamics to conclude that the evolutionary trend is to make the mean delay in growth as short as possible. This confirms Hutchinson's conjecture (Hutchinson in Ann N Y Acad Sci 50(4):221-246, 1948) and fits biological evidence.

Self-complementary circular codes in coding theory.

Self-complementary circular codes are involved in pairing genetic processes. A maximal [Formula: see text] self-complementary circular code X of trinucleotides was identified in genes of bacteria, archaea, eukaryotes, plasmids and viruses (Michel in Life 7(20):1-16 2017, J Theor Biol 380:156-177, 2015; Arquès and Michel in J Theor Biol 182:45-58 1996). In this paper, self-complementary circular codes are investigated using the graph theory approach recently formulated in Fimmel etal. (Philos Trans R Soc A 374:20150058, 2016). A directed graph [Formula: see text] associated with any code X mirrors the properties of the code. In the present paper, we demonstrate a necessary condition for the self-complementarity of an arbitrary code X in terms of the graph theory. The same condition has been proven to be sufficient for codes which are circular and of large size [Formula: see text] trinucleotides, in particular for maximal circular codes ([Formula: see text] trinucleotides). For codes of small-size [Formula: see text] trinucleotides, some very rare counterexamples have been constructed. Furthermore, the length and the structure of the longest paths in the graphs associated with the self-complementary circular codes are investigated. It has been proven that the longest paths in such graphs determine the reading frame for the self-complementary circular codes. By applying this result, the reading frame in any arbitrary sequence of trinucleotides is retrieved after at most 15 nucleotides, i.e., 5 consecutive trinucleotides, from the circular code X identified in genes. Thus, an X motif of a length of at least 15 nucleotides in an arbitrary sequence of trinucleotides (not necessarily all of them belonging to X) uniquely defines the reading (correct) frame, an important criterion for analyzing the X motifs in genes in the future.

Constrained minimization problems for the reproduction number in meta-population models.

The basic reproduction number ([Formula: see text]) can be considerably higher in an SIR model with heterogeneous mixing compared to that from a corresponding model with homogeneous mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (Lancet Infect Dis 16(5):599-605, 2016. https://doi.org/10.1016/S1473-3099(16)00004-9 ), reported an increase of 70% in [Formula: see text] when heterogeneity was accounted for. Meta-population models with simple heterogeneous mixing functions, e.g., proportionate mixing, have been employed to identify optimal vaccination strategies using an approach based on the gradient of the effective reproduction number ([Formula: see text]), which consists of partial derivatives of [Formula: see text] with respect to the proportions immune [Formula: see text] in sub-groups i (Feng et al. inJ Theor Biol 386:177-187,2015. https://doi.org/10.1016/j.jtbi.2015.09.006 ; Math Biosci 287:93-104,2017. https://doi.org/10.1016/j.mbs.2016.09.013 ). These papers consider cases in which an optimal vaccination strategy exists. However, in general, the optimal solution identified using the gradient may not be feasible for some parameter values (i.e., vaccination coverages outside the unit interval). In this paper, we derive the analytic conditions under which the optimal solution is feasible. Explicit expressions for the optimal solutions in the case of [Formula: see text] sub-populations are obtained, and the bounds for optimal solutions are derived for [Formula: see text] sub-populations. This is done for general mixing functions and examples of proportionate and preferential mixing are presented. Of special significance is the result that for general mixing schemes, both [Formula: see text] and [Formula: see text] are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

Eliminating Warrantless Assumptions Facilitates Consideration of an Electrostatic Model of Ion-Channel Activation

Models of voltage-sensitive ion channels (VSICs) that don’t specify the way reduction in electric field at threshold activation leads to observed outward S4 motions don’t provide satisfactory explanations of voltage sensing [Bezanilla F (2008) Neuron50:456-468]. Current models depend on unrealistic assumptions: screening of charges by a putative water-filled pore, and simple mechanical devices. A gate, screw, paddle or other everyday device cannot physically explain activation of the highly energetic resting channel, a far-from-equilibrium macromolecular system: The α-helix's hydrogen bonds are too weak to support such rigid structures. The energy required to lengthen the H-bonds is provided when the capacitive energy lost in the voltage reduction to threshold does mechanical work on the channel. Similarities exist between excitable membranes and ferroelectrics [Leuchtag HR (1987) J Theor Biol 127:321-340; 127:341-359; (2008) Voltage-Sensitive Ion Channels, Springer]. The ferroelectricity of excitable membranes must originate in the channels, since VSICs are their active components. In ferroelectrics, the dielectric coefficient ε is a function of the electric field. Electrostatic forces are inversely proportional to ε. Evidence from molecules with the sidechains of the branched-chain amino acids isoleucine, leucine and valine shows that their ε becomes remarkably large in a high electric field [Yoshino K, Sakurai T (1991) in: Goodby JW et al. Ferroelectric Liquid Crystals:317-363], comparable to that in a resting channel. According to the Channel Activation by Electrostatic Repulsion (CAbER) model, ε is high at rest potential and drops on depolarization. The consequently increased electrostatic repulsion between positively charged arginine and lysine residues at activation expands the S4 segments. This outward motion reconfigures the VSIC into an “open” structure with pore-domain sites wide enough to accommodate unhydrated permeant ions that hop stochastically across the channel [Leuchtag HR (2016) Biophys. J. 112(3):544a; 112(3):543a-544a].

Modeling the Transfer of Drug Resistance in Solid Tumors.

ABC efflux transporters are a key factor leading to multidrug resistance in cancer. Overexpression of these transporters significantly decreases the efficacy of anti-cancer drugs. Along with selection and induction, drug resistance may be transferred between cells, which is the focus of this paper. Specifically, we consider the intercellular transfer of P-glycoprotein (P-gp), a well-known ABC transporter that was shown to confer resistance to many common chemotherapeutic drugs. In a recent paper, Durán et al. (Bull Math Biol 78(6):1218-1237, 2016) studied the dynamics of mixed cultures of resistant and sensitive NCI-H460 (human non-small lung cancer) cell lines. As expected, the experimental data showed a gradual increase in the percentage of resistance cells and a decrease in the percentage of sensitive cells. The experimental work was accompanied with a mathematical model that assumed P-gp transfer from resistant cells to sensitive cells, rendering them temporarily resistant. The mathematical model provided a reasonable fit to the experimental data. In this paper, we develop a new mathematical model for the transfer of drug resistance between cancer cells. Our model is based on incorporating a resistance phenotype into a model of cancer growth (Greene etal. in J Theor Biol 367:262-277, 2015). The resulting model for P-gp transfer, written as a system of integro-differential equations, follows the dynamics of proliferating, quiescent, and apoptotic cells, with a varying resistance phenotype. We show that this model provides a good match to the dynamics of the experimental data of Durán etal. (2016). The mathematical model shows a better fit when resistant cancer cells have a slower division rate than the sensitive cells.

What can be observed in real time PCR and when does it show?

Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299–304, 2003. doi:10.1016/S0022-5193(03)00166-8), and based on Michaelis–Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433–440, 1997). A crucial role is played by the Michaelis–Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when v<1, and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a “veil of uncertainty”. This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.

Strong Comma-Free Codes in Genetic Information.

Comma-free codes constitute a class of circular codes, which has been widely studied, in particular by Golomb et al. (Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab 23:1-34, 1958a, Can J Math 10:202-209, 1958b), Michel et al. (Comput Math Appl 55:989-996, 2008a, Theor Comput Sci 401:17-26, 2008b, Inf Comput 212:55-63, 2012), Michel and Pirillo (Int J Comb 2011:659567, 2011), and Fimmel and Strüngmann (J Theor Biol 389:206-213, 2016). Based on a recent approach using graph theory to study circular codes Fimmel et al. (Philos Trans R Soc 374:20150058, 2016), a new class of circular codes, called strong comma-free codes, is identified. These codes detect a frameshift during the translation process immediately after a reading window of at most two nucleotides. We describe several combinatorial properties of strong comma-free codes: enumeration, maximality, self-complementarity and [Formula: see text]-property (comma-free property in all the three possible frames). These combinatorial results also highlight some new properties of the genetic code and its evolution. Each amino acid in the standard genetic code is coded by at least one strong comma-free code of size 1. There are 9 amino acids [Formula: see text] among 20 such that for each amino acid from S, its synonymous trinucleotide set (excluding the necessary periodic trinucleotides [Formula: see text]) is a strong comma-free code. The primeval comma-free RNY code of Eigen and Schuster (Naturwissenschaften 65:341-369, 1978) is a self-complementary [Formula: see text]-code of size 16. Furthermore, it is the union of two strong comma-free codes of size 8 which are complementary to each other.

A random effects model for the identification of differential splicing (REIDS) using exon and HTA arrays

BackgroundAlternative gene splicing is a common phenomenon in which a single gene gives rise to multiple transcript isoforms. The process is strictly guided and involves a multitude of proteins and regulatory complexes. Unfortunately, aberrant splicing events do occur which have been linked to genetic disorders, such as several types of cancer and neurodegenerative diseases (Fan et al., Theor Biol Med Model 3:19, 2006). Therefore, understanding the mechanism of alternative splicing and identifying the difference in splicing events between diseased and healthy tissue is crucial in biomedical research with the potential of applications in personalized medicine as well as in drug development.ResultsWe propose a linear mixed model, Random Effects for the Identification of Differential Splicing (REIDS), for the identification of alternative splicing events. Based on a set of scores, an exon score and an array score, a decision regarding alternative splicing can be made. The model enables the ability to distinguish a differential expressed gene from a differential spliced exon. The proposed model was applied to three case studies concerning both exon and HTA arrays.ConclusionThe REIDS model provides a work flow for the identification of alternative splicing events relying on the established linear mixed model. The model can be applied to different types of arrays.

Stability for a New Discrete Ratio-Dependent Predator\u2013Prey System

The stability of a new two-species discrete ratio-dependent predator–prey system is considered. By using the linearization method, we obtain some sufficient conditions for the local stability of the positive equilibria. We also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper (Chen and Zhou in J Math Anal Appl 27:7358–7366, 2003) has done. The method given in this paper is new and very resultful comparing with articles (Damgaard in J Theor Biol 227:197–203, 2004; Edmunds in Theor Popul Biol 72:379–388, 2007; Fan and Wang in Math Comput Model 35:951–961, 2002; Muroya in J Math Anal Appl 330:24–33, 2007; Huo and Li in Appl Math Comput 153:337–351, 2004; Liao et al. in Appl Math Comput 190:500–509, 2007) and it can also be applied to study other global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present two open questions.

Mathematical Model of Bone Remodeling Captures the Antiresorptive and Anabolic Actions of Various Therapies.

A better understanding of the molecular pathways regulating the bone remodeling process should help in the development of new antiresorptive regulators and anabolic regulators, that is, regulators of bone resorption and of bone formation. Understanding the mechanisms by which parathyroid hormone (PTH) influences bone formation and how it switches from anabolic to catabolic action is important for treating osteoporosis (Poole and Reeve in Curr Opin Pharmacol 5:612-617, 2005). In this paper we describe a mathematical model of bone remodeling that incorporates, extends, and integrates several models of particular aspects of this biochemical system (Cabal et al. in J Bone Miner Res 28(8):1830-1836, 2013; Lemaire et al. in J Theor Biol 229:293-309, 2004; Peterson and Riggs in Bone 46:49-63, 2010; Raposo et al. in J Clin Endocrinol Metab 87(9):4330-4340, 2002; Ross et al. in J Disc Cont Dyn Sys Series B 17(6):2185-2200, 2012). We plan to use this model as a bone homeostasis platform to develop anabolic and antiresorptive compounds. The model will allow us to test hypotheses about the dynamics of compounds and to test the potential benefits of combination therapies. At the core of the model is the idealized account of osteoclast and osteoblast signaling given by Lemaire et al. (J Theor Biol 229:293-309, 2004). We have relaxed some of their assumptions about the roles of osteoprotegerin, transforming growth factor [Formula: see text], and receptor activator of nuclear factor [Formula: see text]B ligand; we have devised more detailed models of the interactions of these species. We have incorporated a model of the effect of calcium sensing receptor antagonists on remodeling (Cabal et al. in J Bone Miner Res 28(8):1830-1836, 2013). We have also incorporated a basic model of the effects of vitamin D on calcium homeostasis. We have included a simple model of the mechanism proposed by Bellido et al. (2003), Ross et al. (J Disc Cont Dyn Sys Series B 17(6):2185-2200, 2012), of the influence of PTH on osteoblast apoptosis, a mechanism that accounts for the anabolic response to pulsatile PTH administration. Finally, we have devised a simple model of the administration and effects of bisphosphonates. The biomarkers in the model are procollagen type 1 amino-terminal propeptide and C-terminal telopeptide. Bone mineral density is the model's principal endpoint.

A quantitative report on the impact of chloride on the kinetic coefficients of auxin-induced growth: a numerical contribution to the "acid growth hypothesis".

This work presents the application of several our own novel methods of analysing the kinetics of plant growth, which create, among others, a common platform for the comparison of experimental results. A relatively simple formula is used to parameterize the wide range of data that has been obtained for Zea mays L. in the literature, though it can also be used for different species. A biophysical/biochemical interpretation of the parameters was obtained from a theoretical model that is based on a modified Lockhart equation. The derived formula, which was extended for practical use in Zajdel et al. (Acta Physiol Plant 38:5, 2016), and which was implemented in the attached computer program (ibid.), allowed the data that was obtained from the growth-related problems to be parameterized in a simple way. As a working example that shows the robustness of our approach, we comment in detail on the qualitative assessments of the impact of chloride ions on auxin-induced growth. We note that calculated continuous curves (fits), which are rooted in the growth functional that was introduced by Pietruszka (J Theor Biol 315:119–127, 2012), were in a perfect agreement (R2 ~ 0.99998) with the raw experimental data that was published recently by Burdach et al. (Ann Bot 114:1023–1034, 2014). This fact justified the use of this strict technique, which allows for the determination of kinetic coefficients, to critically evaluate the results and suppositions (claims) therein. Moreover, we calculated the time-delay derivative of elongation growth—pH cross-correlations, and validated the “acid growth hypothesis” in figures by considering, amongst others, the magnitude of the H+-activity of elongation growth (per μm). An empirical constant (field strength), EH+ = Em/(log10 1/aH+ ∙ μm) = 0.157 ± 0.009 [V/mm] was obtained, where Em [mV] is the membrane potential in the perenchymal coleoptile cells of Zea mays L. When this relation is known, the membrane potential can not only be determined for intact growth, but also for different intervening substances exclusively from growth (or growth rate) and pH measurements, i.e. without performing electrophysiological measurements. However, the question of whether this constant is universal remains open.Electronic supplementary materialThe online version of this article (doi:10.1186/s40064-016-3626-y) contains supplementary material, which is available to authorized users.

PI-FLAME: A parallel immune system simulator using the FLAME graphic processing unit environment

Agent-based models (ABMs) are increasingly being used to study population dynamics in complex systems, such as the human immune system. Previously, Folcik et al. (The basic immune simulator: an agent-based model to study the interactions between innate and adaptive immunity. Theor Biol Med Model 2007; 4: 39) developed a Basic Immune Simulator (BIS) and implemented it using the Recursive Porous Agent Simulation Toolkit (RePast) ABM simulation framework. However, frameworks such as RePast are designed to execute serially on central processing units and therefore cannot efficiently handle large model sizes. In this paper, we report on our implementation of the BIS using FLAME GPU, a parallel computing ABM simulator designed to execute on graphics processing units. To benchmark our implementation, we simulate the response of the immune system to a viral infection of generic tissue cells. We compared our results with those obtained from the original RePast implementation for statistical accuracy. We observe that our implementation has a 13× performance advantage over the original RePast implementation.

Evolution of Site-Selection Stabilizes Population Dynamics, Promotes Even Distribution of Individuals, and Occasionally Causes Evolutionary Suicide

Species that compete for access to or use of sites, such as parasitic mites attaching to honey bees or apple maggots laying eggs in fruits, can potentially increase their fitness by carefully selecting sites at which they face little or no competition. Here, we systematically investigate the evolution of site-selection strategies among animals competing for discrete sites. By developing and analyzing a mechanistic and population-dynamical model of site selection in which searching individuals encounter sites sequentially and can choose to accept or continue to search based on how many conspecifics are already there, we give a complete characterization of the different site-selection strategies that can evolve. We find that evolution of site-selection stabilizes population dynamics, promotes even distribution of individuals among sites, and occasionally causes evolutionary suicide. We also discuss the broader implications of our findings and propose how they can be reconciled with an earlier study (Nonaka et al. in J Theor Biol 317:96–104, 2013) that reported selection toward ever higher levels of aggregation among sites as a consequence of site-selection.

Dependence of the structure and mechanics of metaphase chromosomes on oxidized cysteines.

We have found that reagents that reduce oxidized cysteines lead to destabilization of metaphase chromosome folding, suggesting that chemically linked cysteine residues may play a structural role in mitotic chromosome organization, in accord with classical studies by Dounce et al. (J Theor Biol 42:275-285, 1973) and Sumner (J Cell Sci 70:177-188, 1984a). Human chromosomes isolated into buffer unfold when exposed to dithiothreitol (DTT) or tris(2-carboxyethyl)phosphine (TCEP). In micromanipulation experiments which allow us to examine the mechanics of individual metaphase chromosomes, we have found that the gel-like elastic stiffness of native metaphase chromosomes is dramatically suppressed by DTT and TCEP, even before the chromosomes become appreciably unfolded. We also report protein labeling experiments on human metaphase chromosomes which allow us to tag oxidized and reduction-sensitive cysteine residues. PAGE analysis using fluorescent labels shows a small number of labeled bands. Mass spectrometry analysis of similarly labeled proteins provides a list of candidates for proteins with oxidized cysteines involved in chromosome organization, notably including components of condensin I, cohesin, the nucleosome-interacting proteins RCC1 and RCC2, as well as the RNA/DNA-binding protein NONO/p54NRB.

Numerical simulations of a reduced model for blood coagulation

In this work, the three-dimensional numerical resolution of a complex mathematical model for the blood coagulation process is presented. The model was illustrated in Fasano et al. (Clin Hemorheol Microcirc 51:1–14, 2012), Pavlova et al. (Theor Biol 380:367–379, 2015). It incorporates the action of the biochemical and cellular components of blood as well as the effects of the flow. The model is characterized by a reduction in the biochemical network and considers the impact of the blood slip at the vessel wall. Numerical results showing the capacity of the model to predict different perturbations in the hemostatic system are discussed.

A new modified resource budget model for nonlinear dynamics in citrus production

Alternate bearing or masting is a general yield variability phenomenon in perennial tree crops. This paper first presents a theoretical modeling and simulation study of the mechanism for this dynamics in citrus, and then provides a test of the proposed models using data from a previous 16-year experiment in a citrus orchard. Our previous studies suggest that the mutual effects between vegetative and reproductive growths caused by resource allocation and budgeting in plant body might be considered as a major factor responsible for the yield oscillations in citrus. Based on the resource budget model proposed by Isagi et al. (J Theor Biol. 1997;187:231-9), we first introduce the new leaf growth as a major energy consumption component into the model. Further, we introduce a nonlinear Ricker-type equation to replace the linear relationship between costs for flowering and fruiting used in Isagi's model. Model simulations demonstrate that the proposed new models can successfully simulate the reproductive behaviors of citrus trees with different fruiting dynamics. These results may enrich the mechanical dynamics in tree crop reproductive models and help us to better understand the dynamics of vegetative-reproductive growth interactions in a real environment.

The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle.

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin’s maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.