A model for the spatial transmission of dengue with daily movement between villages and a city.

Dengue is a re-emergent vector-borne disease affecting large portions of the world's population living in the tropics and subtropics. The virus is transmitted through the bites of female Aedes aegypti mosquitoes, and it is widely believed that these bites occur primarily in the daytime. The transmission of dengue is a complicated process, and one of the main sources of this complexity is due to the movement of people, e.g. between home and their places of work. Hence, the mechanics of disease progression may also differ between day and night. A discrete-time multi-patch dengue transmission model which takes into account the mobility of people as well as processes of infection, recovery, recruitment, mortality, and outbound and return movements is considered here. One patch (the city) is connected to all other patches (the villages) in a spoke-like network. We obtain here the basic reproductive ratio (ℛ0) of the transmission model which represents a threshold for an epidemic to occur. Dynamical analysis for vector control, human treatment and vaccination, and different kinds of mobility are performed. It is shown that changes in human movement patterns can, in some situations, affect the ability of the disease to persist in a predictable manner. We conclude with biological implications for the prevention and control of dengue virus transmission.

Effect of resource subsidies on predator-prey population dynamics: a mathematical model.

The influence of a resource subsidy on predator-prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator-prey-subsidy interactions also increases.

Retention time variability as a mechanism for animal mediated long-distance dispersal.

Long-distance dispersal (LDD) events, although rare for most plant species, can strongly influence population and community dynamics. Animals function as a key biotic vector of seeds and thus, a mechanistic and quantitative understanding of how individual animal behaviors scale to dispersal patterns at different spatial scales is a question of critical importance from both basic and applied perspectives. Using a diffusion-theory based analytical approach for a wide range of animal movement and seed transportation patterns, we show that the scale (a measure of local dispersal) of the seed dispersal kernel increases with the organisms' rate of movement and mean seed retention time. We reveal that variations in seed retention time is a key determinant of various measures of LDD such as kurtosis (or shape) of the kernel, thinkness of tails and the absolute number of seeds falling beyond a threshold distance. Using empirical data sets of frugivores, we illustrate the importance of variability in retention times for predicting the key disperser species that influence LDD. Our study makes testable predictions linking animal movement behaviors and gut retention times to dispersal patterns and, more generally, highlights the potential importance of animal behavioral variability for the LDD of seeds.

Stability of choice in the honey bee nest-site selection process.

We introduce a pair of compartment models for the honey bee nest-site selection process that lend themselves to analytic methods. The first model represents a swarm of bees deciding whether a site is viable, and the second characterizes its ability to select between two viable sites. We find that the one-site assessment process has two equilibrium states: a disinterested equilibrium (DE) in which the bees show no interest in the site and an interested equilibrium (IE) in which bees show interest. In analogy with epidemic models, we define basic and absolute recruitment numbers (R(0) and B(0)) as measures of the swarm's sensitivity to dancing by a single bee. If R(0) is less than one then the DE is locally stable, and if B(0) is less than one then it is globally stable. If R(0) is greater than one then the DE is unstable and the IE is stable under realistic conditions. In addition, there exists a critical site quality threshold Q(*) above which the site can attract some interest (at equilibrium) and below which it cannot. We also find the existence of a second critical site quality threshold Q(**) above which the site can attract a quorum (at equilibrium) and below which it cannot. The two-site discrimination process, in which we examine a swarm's ability to simultaneously consider two sites differing in both site quality and discovery time, has a stable DE if and only if both sites' individual basic recruitment numbers are less than one. Numerical experiments are performed to study the influences of site quality on quorum time and the outcome of competition between a lower quality site discovered first and a higher quality site discovered second.

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.

The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model.

A global method of nullcline endpoint analysis is employed to determine the outcome of competition for sunlight between two hypothetical plant species with clonal growth form that differ solely in the height at which they place their leaves above the ground. This difference in vertical leaf placement, or canopy partitioning, produces species differences in sunlight energy capture and stem metabolic maintenance costs. The competitive interaction between these two species is analyzed by considering a special case of a canopy partitioning model (RR Vance and AL Nevai, J. Theor. Biol. 2007, 245:210-219; AL Nevai and RR Vance, J. Math. Biol. 2007, 55:105-145). Nullcline endpoint analysis is used to partition parameter space into regions within which either competitive exclusion or competitive coexistence occurs. The principal conclusion is that two clonal plant species which compete for sunlight and place their leaves at different heights above the ground but differ in no other way can, under suitable parameter values, experience stable coexistence even though they occupy an environment which varies neither over horizontal space nor through time.

State-dependent choice and ecological rationality.

Decision makers who minimize costly errors should flexibly adjust the way they trade off competing demands, depending on their current state. We explore how state (amount of hoarded food) affects willingness to take extra predation risk to obtain larger food rewards, particularly in animals that may overemphasize safety. Assuming a sigmoid fitness function, we explore how a supplement in state influences this willingness trade danger for food energy. Above a threshold, the model predicts the supplement will weaken this willingness. Incremental increases in state in the deceleratory phase yield smaller fitness gains, so it pays to increase emphasis on safety after receiving a supplement. Below this threshold, the model makes the opposite prediction because incremental increases in state yield bigger fitness gains and so it pays to decrease emphasis on safety. We use the model to explain why hoarding gray jays (Perisoreus canadensis) were induced by an experimental subsidy to accept greater danger. This formerly puzzling finding makes sense if the jays' effective hoard was relatively small, due to theft and decomposition. We discuss adaptive state-dependent choice as a general explanation for apparently irrational behavior.

Plant interspecies competition for sunlight: a mathematical model of canopy partitioning.

We examine the influence of canopy partitioning on the outcome of competition between two plant species that interact only by mutually shading each other. This analysis is based on a Kolmogorov-type canopy partitioning model for plant species with clonal growth form and fixed vertical leaf profiles (Vance and Nevai in J. Theor. Biol., 2007, to appear). We show that canopy partitioning is necessary for the stable coexistence of the two competing plant species. We also use implicit methods to show that, under certain conditions, the species' nullclines can intersect at most once. We use nullcline endpoint analysis to show that when the nullclines do intersect, and in such a way that they cross, then the resulting equilibrium point is always stable. We also construct surfaces that divide parameter space into regions within which the various outcomes of competition occur, and then study parameter dependence in the locations of these surfaces. The analysis presented here and in a companion paper (Nevai and Vance, The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model, in review) together shows that canopy partitioning is both necessary and, under appropriate parameter values, sufficient for the stable coexistence of two hypothetical plant species whose structure and growth are described by our model.

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.