Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning

Can a difference in the heights at which plants place their leaves, a pattern we call canopy partitioning, make it possible for two competing plant species to coexist? To find out, we examine a model of clonal plants living in a nonseasonal environment that relates the dynamical behavior and competitive abilities of plant populations to the structural and functional features of the plants that form them. This examination emphasizes whole plant performance in the vertical light gradient caused by self-shading. This first of three related papers formulates a prototype single species Canopy Structure Model from biological first principles and shows how all plant properties work together to determine population persistence and equilibrium abundance. Population persistence is favored, and equilibrium abundance is increased, by high irradiance, high maximum photosynthesis rate, rapid saturation of the photosynthetic response to increased irradiance, low tissue respiration rate, small amounts of stem and root tissue necessary to support the needs of leaves, and low density of leaf, stem, and root tissues. In particular, equilibrium abundance decreases as mean leaf height increases because of the increased cost of manufacturing and maintaining stem tissue. All conclusions arise from this formulation by straightforward analysis. The argument concludes by stating this formulation's straightforward extension, called a Canopy Partitioning Model, to two competing species.     

Plant competition for light analyzed with a multispecies canopy model

Summary A multispecies canopy photosynthesis simulation model was used to examine the importance of canopy structure in influencing light interception and carbon gain in mixed and pure stands of wheat (Triticum aestivum L.) and wild oat (Avena fatua L.), a common weedy competitor of wheat. In the mixtures, the fraction of the simulated canopy photosynthesis contributed by wheat was found to decline during the growing season and this decline was closely related to reductions in the amount of leaf area in upper canopy layers. For both species in mixture and in monoculture, simulated photosynthesis was greatest in the middle or upper-middle canopy layers and sensitivity analyses revealed that canopy photosynthesis was most sensitive to changes in leaf area and leaf inclination in these layers. Changes in LAI and leaf inclination affected canopy carbon gain differently for mixtures and monocultures, but the responses were not the same for the two species. Results from simulations where the structural characteristics of the two species were substituted indicated that species differences in leaf inclination, sheath area and the fraction of leaf area alive were of minor consequence compared with the differences in total leaf area in influencing relative canopy carbon gain in mixtures. Competition for light in these species mixtures appears to be influenced most by differences in the positioning of leaf area in upper canopy layers which determines, to a great extent, the amount of light intercepted.     

Plant competition for light analyzed with a multispecies canopy model

Summary Competition for light among species in a mixed canopy can be assessed quantitatively by a simulation model which evaluates the importance of different morphological and photosynthetic characteristics of each species. A model was developed that simulates how the foliage of all species attenuate radiation in the canopy and how much radiation is received by foliage of each species. The model can account for different kinds of foliage (leaf blades, stems, etc.) for each species. The photosynthesis and transpiration for sunlit and shaded foliage of each species is also computed for different layers in the canopy. The model is an extension of previously described single-species canopy photosynthesis simulation models. Model predictions of the fraction of foliage sunlit and interception of light by sunlit and shaded foliage for monoculture and mixed canopies of wheat (Triticum aestivum) and wild oat (Avena fatua) in the field compared very well with measured values. The model was used to calculate light interception and canopy photosynthesis for both species of wheat/wild oat mixtures grown under normal solar and enhanced ultraviolet-B (290–320 nm) radiation (UV-B) in a glasshouse experiment with no root competition. In these experiments, measurements showed that the mixtures receiving enhanced UV-B radiation had a greater proportion of the total foliage area composed of wheat compared to mixtures in the control treatments. The difference in species foliage area and its position in the canopy resulted in a calculated increase in the portion of total canopy radiation interception and photosynthesis by wheat. This, in turn, is consistent with greater canopy biomass of wheat reported in canopies irradiated with supplemental UV-B.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210–219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine “interior” and “exterior” species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Photosynthetic capacity and nitrogen partitioning among species in the canopy of a herbaceous plant community

Partitioning of nitrogen among species was determined in a stand of a tall herbaceous community. Total amount of nitrogen in the aboveground biomass was 261 mmol N m^–2, of which 92% was in three dominant species (Phragmites, Calamagrostis and Carex) and the rest was in the other eight subordinate species. Higher nitrogen concentrations per unit leaf area (n _L) with increasing photosynthetically active photon flux density (PPFD) were observed in all species except for three short species. The changes in n _L within species were mainly explained by the different nitrogen concentrations per unit leaf mass, while the differences in n _L between species were explained by the different SLM (leaf mass per unit leaf area). Photon absorption per unit leaf nitrogen ( _N ) was determined for each species. If photosynthetic activity was proportional to photon absorption, _N should indicate in situ PNUE (photosynthetic nitrogen use efficiency). High _N of Calamagrostis (dominant) resulted from high photon absorption per unit leaf area ( _area ), whereas high _N of Scutellaria (subordinate) resulted from low n _L although its _area was low. Species with cylinder-like leaves (Juncus and Equisetum) had low _N , which resulted from their high n _L. Light-saturated CO₂ exchange rates per unit leaf area (CER) and per unit leaf nitrogen (potential PNUE) were determined in seven species. Species with high CER and high n _L (Phragmites, Carex and Juncus) had low potential PNUE, while species with low CER and low n _L showed high potential PNUE. NUE (ratio of dry mass production to nitrogen uptake) was approximated as a reciprocal of plant nitrogen concentration. In most species, three measures of nitrogen use efficiency (NUE, _N and potential PNUE) showed strong conformity. Nitrogen use efficiency was high in Calamagrostis and Scutellaria, intermediate in Phragmites and relatively low in Carex. Nitrogen use efficiency of subordinate species was as high as or even higher than that of dominant species, which suggests that growth is co-limited by light and nitrogen in the subordinate species.     

A mathematical model for water and nutrient uptake by plant root systems

This article deals with modelling the simultaneous uptake of water and highly buffered nutrient, such as phosphate, by root branching structures from partially saturated soil. We use the simultaneous water and nutrient uptake model to investigate the effect that water movement has on nutrient uptake. With the aid of this model we are also able to show that the previous models by Barber and Tinker and Nye systematically underestimated the phosphate uptake, due to the oversimplified approach in dealing with root branching structure. In this article we show how this discrepancy can be remedied and the root branching structure included in the models of plant nutrient uptake. We will also discuss the differences in the results for continuous and spot fertilization combined with variable rainfall.     

Addition of a non-photic component to a light-based mathematical model of the human circadian pacemaker

Mathematical models have become vital to the study of many biological processes in humans due to the complexity of the physiological mechanisms underlying these processes and systems. While our current mathematical representation of the human circadian pacemaker has proven useful in many experimental situations, it uses as input only a direct effect of light on the circadian pacemaker. Although light (a photic stimulus) has been shown to be the primary synchronizer of the circadian pacemaker across a number of species, studies in both animals and humans have confirmed the existence of non-photic effects that also contribute to phase shifting and entrainment. We modified our light-based circadian mathematical model to reflect evidence from these studies that the sleep-wake cycle and/or associated behaviors have a non-photic effect on the circadian pacemaker. In our representation, the sleep-wake cycle and its associated behaviors provides a non-photic drive on the circadian pacemaker that acts both independently and concomitantly with light stimuli. Further experiments are required to validate fully our model and to understand the exact effect of the sleep-wake cycle as a non-photic stimulus for the human circadian pacemaker.     

Canopy structure,light microclimate and leaf gas exchange of Quercus coccifera L. in a Portuguese macchia: measurements in different canopy layers and simulations with a canopy model

Summary The structural characteristics of a diverse array of Quercus coccifera canopies were assessed and related to measured and computed light attenuation, proportion of sunlit foliage, foliage temperatures, and photosynthesis and diffusive conductance behavior in different canopy layers. A canopy model incorporating all components of shortwave and longwave radiation, and the energy balance, conductance, and CO₂ and H₂O exchanges of all leaf layers was developed and compared with measurements of microclimate and gas exchange in canopies in four seasons of the year. In the denser canopies with a leaf area index (LAI) greater than 5, there is little sunlit foliage and the diffuse radiation (400–700 nm) is attenuated to 5% or less of the global radiation (400–700 nm) incident on the top of the canopy. Foliage of this species is nonrandomly distributed with respect to azimuth angle, and within each canopy layer, foliage azimuth and inclination angles are correlated. A detailed version of the model which computed radiation interception and photosynthetic light harvesting according to these nonrandom distributions indicated little difference in whole-canopy gas exchange from calculations of the normal model, which assumes random azimuth orientation. The contributions of different leaf layers to canopy gas exchange are not only a function of the canopy microclimate, but also the degree to which leaves in the lower layers of the canopy exhibit more shade-leaf characteristics, such as low photosynthetic and respiratory capacity and maximal conductance. On cloudless days, the majority of the foliage in a canopy of 5.4 LAI is shaded —70%–90% depending on the time of year. Yet, the shaded foliage under these conditions is calculated to contribute only about one-third of the canopy carbon gain. This contribution is about the same as that of the upper 13% of the canopy foliage. Computed annual whole-canopy carbon gain and water use are, respectively, 60% and 100% greater for a canopy of 5 LAI than for one of 2 LAI. Canopy water-use efficiency is correspondingly less for the canopy of 5 LAI than for that of 2 LAI, but most of this difference is apparent during the cool months of the year, when moisture is more abundant.     

Sunfleck dynamics and canopy structure in a Phaseolus vulgaris L. canopy

 Photosynthetic photon flux density (PPFD) fluctuations were quantified in crops of beans (Phaseolus vulgaris L.) in the field as the canopy developed between July and October. Two different methods were used to select sunflecks and shadeflecks. Four ranges of zenith angles (60–70°, 50–60°, 40–50° and 30–40°) were selected for analysing PPFD fluctuations. At the base of the canopy, sunflecks contributed 18%, 53%, 10% and 4% during the 1st, 3rd, 5th and 7th week of growth, respectively. At a height of 20 cm above the soil surface, the respective contributions were 28% and 21% during the 6th and 7th weeks. Sunfleck lengths of 0–5 s were the most frequent, with the greatest number being found with smaller zenith angles. The proportion of short duration sunflecks increased as the growth period advanced. The number of long sunflecks decreased with time, with very few longer than 100 s by the 5th and 7th weeks. The distributions of sunfleck irradiance were similar to normal distributions and irradiance ranged in μmol m⁻² s⁻¹ from 600–900, 800–1500 and 1000–1600 respectively at zenith angles of 50–60°, 40–50° and 30–40°. A multiple regression showed that short sunflecks (<100 s) depended on zenith angle, plant height, and leaf and stem area index (L _s ), whereas long sunflecks (>100 s) depended on zenith angle and L _s. Shadefleck distributions were similar to those for sunflecks but there were fewer of the shortest examples and more of the longest. The best statistical distribution to describe sunflecks and shadeflecks was the gamma distribution, which could provide the basis for the future development of a good model for sunfleck and shadefleck distributions. Received: 10 November 1997 / Accepted: 2 April 1998     

Requirements for plant coexistence through pollination niche partitioning

Plant–pollinator interactions are often thought to have been a decisive factor in the diversification of flowering plants, but to be of little or no importance for the maintenance of existing plant diversity. In a recent opinion paper, Pauw (2013 Trends Ecol. Evol. 28, 30–37. (doi:10.1016/j.tree.2012.07.019)) challenged this view by proposing a mechanism of diversity maintenance based on pollination niche partitioning. In this article, I investigate under which conditions the mechanism suggested by Pauw can promote plant coexistence, using a mathematical model of plant and pollinator population dynamics. Numerical simulations show that this mechanism is most effective when the costs of searching for flowers are low, pollinator populations are strongly limited by resources other than pollen and nectar, and plant–pollinator interactions are sufficiently specialized. I review the empirical literature on these three requirements, discuss additional factors that may be important for diversity maintenance through pollination niche partitioning, and provide recommendations on how to detect this coexistence mechanism in natural plant communities.     

A mathematical model of hematopoiesis: II. Cyclical neutropenia

Cyclical neutropenia is a dynamical disease of the hematopoietic system marked by an oscillation in circulating leukocyte (e.g. neutrophil) numbers to near zero levels and then back to normal. This oscillation is also mirrored in the platelets and reticulocytes which oscillate with the same period. Cyclical neutropenia has an animal counterpart in the grey collie. Using the mathematical model of the hematopoietic system of Colijn and Mackey [A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. Companion paper to the present paper.] we have determined what parameters are necessary to mimic laboratory and clinical data on untreated grey collies and humans, and also what changes in these parameters are necessary to fit data during treatment with granulocyte colony stimulating factor (G-CSF). Compared to the normal steady-state values, we found that the major parameter changes that mimic untreated cyclical neutropenia correspond to a decreased amplification (increased apoptosis) within the proliferating neutrophil precursor compartment, and a decrease in the maximal rate of re-entry into the proliferative phase of the stem cell compartment. For the data obtained during G-CSF treatment, good fits were obtained only when parameters were altered that would imply that G-CSF led to higher amplification (lower rate of apoptosis) in the proliferating neutrophil precursors, and a elevated rate of differentiation into the neutrophil line.     

The dietary basis for temporal partitioning: food habits of coexisting Acomys species

Two rodent species of the genus Acomys coexist on rocky terrain in the southern deserts of Israel. The common spiny mouse (A. cahirinus) is nocturnally active whereas the golden spiny mouse (A. russatus) is diurnally active. An early removal study suggested that competition accounts for this pattern of temporal partitioning: the golden spiny mouse is forced into diurnal activity by its congener. Theoretically, temporal segregation should facilitate coexistence if the shared limiting resources differ at different times (primarily among predators whose prey populations have activity rhythms), or if they are renewed within the period of the temporal segregation. We studied food preferences of the two Acomys species in a controlled cafeteria experiment in order to assess resource overlap and the potential for competition for food between the two species. We found no significant difference in food preferences between species. The dietary items preferred by both were arthropods. We also carried out a seasonal study of the percentage and identity of arthropods taken in the field by individuals of the two species. Individuals of both species took on annual average a high percentage of arthropods in their diets. Seasonal diet shifts reflect seasonal abundance of arthropods at Ein Gedi during day and night. Diurnal activity may also reduce interspecific interference competition between A. russatus and A. cahirinus. However, the strong interspecific dietary overlap in food preference, the heavy reliance on arthropods in spiny mouse diets, and the seasonal and circadian differences in arthropod consumption suggest that prey partitioning may be a viable mechanism of coexistence in this system. Received: 6 July 1998 / Accepted: 10 May 1999     

Analysis of stand density effects on canopy structure: a conceptual approach

Summary A few assumptions were used to generate a series of specific, quantitative predictions for the relationships between stand density and various dimensional measures of canopy structure. The predictions, each indicating an increase in mean crown size as density decreased, appeared to be reasonable and intuitive. Predictions were compared to data for two conifer species with different crown forms, Pinus contorta var. latifolia and Abies lasiocarpa. Results of these comparisons were mixed — the linear, directly measured dimensions were consistent with predicted relationships, but dimensions calculated from the linear measures were not. Re-examination of the original assumptions indicated that the model should account for crown shyness (engagement/disengagement) to adequately reflect the influence of stand density on canopy structure. The results also indicated a strong association between stand height and measures of mean crown size. Mean crown size of lodgepole pine was altered much more by density than was mean crown size of subalpine fir, due primarily to the different relative shade tolerances of the two species. Some of the observed differences between species may also reflect the range of densities examined and uneven spacing in the unmanaged natural stands.     

Analysis of a mathematical model for tuberculosis: What could be done to increase case detection

This paper presents qualitative and quantitative study of a TB mathematical model to test results from a survey carried out in Benin City, Nigeria. The purpose of the survey was to determine factors that could enhance the case detection rate of tuberculosis. Results from the survey identified four key factors that must be combined for an effective control of TB and increase the case detection rate: effective awareness programme, active cough identification, associated cost factor for treatment of identified cases and effective treatment. The overall effect of these factors on the basic reproduction number under treatment, R_T, of the TB model was considered. In all, a serious concentration on tuberculosis awareness programmes and active cough identification as a marker for someone having TB was shown to significantly reduce the value of the reproduction number, hereby reducing the severity of the disease in the presence of treatment.     

A competition model between Pseudomonas fluorescens and pathogens via iron chelation

In this study we present a competition model between a non-chelator (e.g. pathogen) microorganism and an iron chelator microorganism (e.g. Pseudomonas fluorescens). This latter is a beneficial bacteria that can inhibit the growth of the non-chelator through its iron chelating capability. This phenomena of iron chelation is shown to prevent the pathogen from proliferating to numbers capable of causing disease. A mathematical model is formulated and used to study this competition. The model proposes a new and simple conceptual explanation of interactions. It is a nonlinear system of ordinary differential equations. A qualitative analysis of the model for the batch case (no inflow or outflow from the system) is carried out and the global behavior of the model variables is studied. For the chemostat case, the equilibrium points were derived and their stability was performed through extensive numerical simulations. It is found that iron chelation is able to control the non-chelator microorganism growth under a wide range of conditions.     

Effect of light interception by non-photosynthetic organs on canopy photosynthetic production

A canopy photosynthesis model was derived on the assumption that the light diminution within a canopy is caused by both leaves and non-photosynthetic organs. The light diminution by leaves and that by non-photosynthetic organs were taken into account separately in the Lambert-Beer equation of light extinction. The light flux density on the leaf surface at each depth was evaluated from the leaf's share of light. The light flux density on the leaf surface thus obtained was incorporated into the Monsi-Saeki model of canopy photosynthesis. The proposed model was applied for estimating gross canopy photosynthesis in a 19-year-oldLarix leptolepis plantation where 38% of the light diminution was due to non-photosynthetic organs. The daily canopy photosynthesis on one summer day calculated using the present model was about 22% less than that calculated by the conventional Monsi-Saeki model, in which light interception by non-photosynthetic organs is neglected. The degree of such reduction in canopy photosynthesis through shading by non-photosynthetic organs was assessed in relation to parameters affecting light extinction, leaf photosynthetic characteristics, and light regime above the canopy.     

Influence of viscosity on myocardium mechanical activity: a mathematical model

We have previously proposed and validated a mathematical model of myocardium contraction-relaxation cycle based on current knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28 rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of viscosity to main inotropic and lusitropic characteristics of myocardium performance.     

A mathematical model of canine granulocytopoiesis

Summary The granulocyte cell renewal system of the dog is represented by a mathematical model consisting of the following compartments: The pool of pluripotential stem cells, the committed stem cell pool, divided into a blood and a bone marrow compartment, the proliferation pool, the maturation pool, the reserve pool and the blood pool of functional granulocytes. This chain of compartments is described by a system of non-linear differential equations. Cell losses anyplace in the system provoke increased production in all pools containing cells capable to divide. A reduced number of granulocytes in the blood pool stimulates production of a granulocyte releasing factor which mobilizes a rising number of cells to transit from the marrow reserve into the blood pool.The model was simulated on a digital computer. It was found to be capable to reproduce the steady state conditions and it also fits the data of two distinct experimental perturbations of the system both equally well. These perturbations are a loss of proliferating cells as it occurs after the administration of cytostatic drugs and losses of functional cells as they are induced by leukapheresis experiments of differing leukapheresis rates.This study was supported by the Deutsche Forschungsgemeinschaft (SFB 112)     

A mathematical model of hematopoiesis--I. Periodic chronic myelogenous leukemia

Periodic chronic myelogenous leukemia (PCML) is an interesting dynamical disease of the hematopoietic system in which oscillating levels of circulating leukocytes, platelets and/or reticulocytes are observed. Typically all of these three differentiated cell types have the same oscillation period, but the relation of the oscillation mean and amplitude to the normal levels is variable. Given the appearance of the abnormal Philadelphia chromosome in all of the nucleated progeny of the hematopoietic stem cells (HSCs), the most parsimonious conclusion is that chronic myelogenous leukemia, and its periodic variant, arise from derangements partially involving the dynamics of the stem cells. Here, we have synthesized several previous mathematical models of HSC dynamics, and models for the regulation of neutrophils, platelets and erythrocytes into a comprehensive model for the regulation of the hematopoietic system. Based on estimates of parameters for a typical normal human, we have systematically explored the changes in some of these parameters necessary to account for the quantitative data on leukocyte, platelet and reticulocyte cycling in 11 patients with PCML. Our results indicate that the critical model parameter changes required to simulate the PCML patient data are an increase in the amplification in the leukocyte line, an increase in the differentiation rate from the stem cell compartment into the leukocyte line, and the rate of apoptosis in the stem cell compartment. Our model system is particularly sensitive to changes in stem cell apoptosis rates, suggesting that changes in the numbers of proliferating stem cells may be important in generating PCML.