Identifying priority sites for whale shark ship collision management globally.

The expansion of the world's merchant fleet poses a great threat to the ocean's biodiversity. Collisions between ships and marine megafauna can have population-level consequences for vulnerable species. The Endangered whale shark (Rhincodon typus) shares a circumglobal distribution with this expanding fleet and tracking of movement pathways has shown that large vessel collisions pose a major threat to the species. However, it is not yet known whether they are also at risk within aggregation sites, where up to 400 individuals can gather to feed on seasonal bursts of planktonic productivity. These "constellation" sites are of significant ecological, socio-economic and cultural value. Here, through expert elicitation, we gathered information from most known constellation sites for this species across the world (>50 constellations and >13,000 individual whale sharks). We defined the spatial boundaries of these sites and their overlap with shipping traffic. Sites were then ranked based on relative levels of potential collision danger posed to whale sharks in the area. Our results showed that researchers and resource managers may underestimate the threat posed by large ship collisions due to a lack of direct evidence, such as injuries or witness accounts, which are available for other, sub-lethal threat categories. We found that constellations in the Arabian Sea and adjacent waters, the Gulf of Mexico, the Gulf of California, and Southeast and East Asia, had the greatest level of collision threat. We also identified 39 sites where peaks in shipping activity coincided with peak seasonal occurrences of whale sharks, sometimes across several months. Simulated collision mitigation options estimated potentially minimal impact to industry, as most whale shark core habitat areas were small. Given the threat posed by vessel collisions, a coordinated, multi-national approach to mitigation is needed within priority whale shark habitats to ensure collision protection for the species.

Global collision-risk hotspots of marine traffic and the world's largest fish, the whale shark.

Marine traffic is increasing globally yet collisions with endangered megafauna such as whales, sea turtles, and planktivorous sharks go largely undetected or unreported. Collisions leading to mortality can have population-level consequences for endangered species. Hence, identifying simultaneous space use of megafauna and shipping throughout ranges may reveal as-yet-unknown spatial targets requiring conservation. However, global studies tracking megafauna and shipping occurrences are lacking. Here we combine satellite-tracked movements of the whale shark, Rhincodon typus, and vessel activity to show that 92% of sharks' horizontal space use and nearly 50% of vertical space use overlap with persistent large vessel (>300 gross tons) traffic. Collision-risk estimates correlated with reported whale shark mortality from ship strikes, indicating higher mortality in areas with greatest overlap. Hotspots of potential collision risk were evident in all major oceans, predominantly from overlap with cargo and tanker vessels, and were concentrated in gulf regions, where dense traffic co-occurred with seasonal shark movements. Nearly a third of whale shark hotspots overlapped with the highest collision-risk areas, with the last known locations of tracked sharks coinciding with busier shipping routes more often than expected. Depth-recording tags provided evidence for sinking, likely dead, whale sharks, suggesting substantial "cryptic" lethal ship strikes are possible, which could explain why whale shark population declines continue despite international protection and low fishing-induced mortality. Mitigation measures to reduce ship-strike risk should be considered to conserve this species and other ocean giants that are likely experiencing similar impacts from growing global vessel traffic.

Discrete epidemic models with two time scales.

The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible-exposed-infectious-recovered-susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number R 0 . Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number R ‾ 0 of the reduced system gives a good approximation of R 0 of the full model since it serves at analyzing its asymptotic behavior. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.

Whale sharks increase swimming effort while filter feeding, but appear to maintain high foraging efficiencies.

Whale sharks (Rhincodon typus) - the largest extant fish species - reside in tropical environments, making them an exception to the general rule that animal size increases with latitude. How this largest fish thrives in tropical environments that promote high metabolism but support less robust zooplankton communities has not been sufficiently explained. We used open-source inertial measurement units (IMU) to log 397 h of whale shark behavior in Yucatán, Mexico, at a site of both active feeding and intense wildlife tourism. Here we show that the strategies employed by whale sharks to compensate for the increased drag of an open mouth are similar to ram feeders five orders of magnitude smaller and one order of magnitude larger. Presumed feeding constituted 20% of the total time budget of four sharks, with individual feeding bouts lasting up to 11 consecutive hours. Compared with normal, sub-surface swimming, three sharks increased their stroke rate and amplitude while surface feeding, while one shark that fed at depth did not demonstrate a greatly increased energetic cost. Additionally, based on time-depth budgets, we estimate that aerial surveys of shark populations should consider including a correction factor of 3 to account for the proportion of daylight hours that sharks are not visible at the surface. With foraging bouts generally lasting several hours, interruptions to foraging during critical feeding periods may represent substantial energetic costs to these endangered species, and this study presents baseline data from which management decisions affecting tourist interactions with whale sharks may be made.

Reduction of nonautonomous population dynamics models with two time scales.

The purpose of this work is reviewing some reduction results to deal with systems of nonautonomous ordinary differential equations with two time scales. They could be included among the so-called approximate aggregation methods. The existence of different time scales in a system, together with some long-term features, are used to build up a simpler system governed by a lesser number of state variables. The asymptotic behavior of the latter system is then used to describe the asymptotic behaviour of the former one. The reduction results are stated in two particular but important cases: periodic systems and asymptotically autonomous systems. The reduction results are illustrated with the help of simple spatial SIS epidemic models including either periodic or asymptotically autonomous terms.

Horizontal movements, migration patterns, and population structure of whale sharks in the Gulf of Mexico and northwestern Caribbean sea.

Whale sharks, Rhincodon typus, aggregate by the hundreds in a summer feeding area off the northeastern Yucatan Peninsula, Mexico, where the Gulf of Mexico meets the Caribbean Sea. The aggregation remains in the nutrient-rich waters off Isla Holbox, Isla Contoy and Isla Mujeres, Quintana Roo for several months in the summer and then dissipates between August and October. Little has been known about where these sharks come from or migrate to after they disperse. From 2003-2012, we used conventional visual tags, photo-identification, and satellite tags to characterize the basic population structure and large-scale horizontal movements of whale sharks that come to this feeding area off Mexico. The aggregation comprised sharks ranging 2.5-10.0 m in total length and included juveniles, subadults, and adults of both sexes, with a male-biased sex ratio (72%). Individual sharks remained in the area for an estimated mean duration of 24-33 days with maximum residency up to about 6 months as determined by photo-identification. After leaving the feeding area the sharks showed horizontal movements in multiple directions throughout the Gulf of Mexico basin, the northwestern Caribbean Sea, and the Straits of Florida. Returns of individual sharks to the Quintana Roo feeding area in subsequent years were common, with some animals returning for six consecutive years. One female shark with an estimated total length of 7.5 m moved at least 7,213 km in 150 days, traveling through the northern Caribbean Sea and across the equator to the South Atlantic Ocean where her satellite tag popped up near the Mid-Atlantic Ridge. We hypothesize this journey to the open waters of the Mid-Atlantic was for reproductive purposes but alternative explanations are considered. The broad movements of whale sharks across multiple political boundaries corroborates genetics data supporting gene flow between geographically distinct areas and underscores the need for management and conservation strategies for this species on a global scale.

Stand dynamics and tree coexistence in an analytical structured model: the role of recruitment.

Understanding the mechanisms of coexistence and niche partitioning in plant communities is a central question in ecology. Current theories of forest dynamics range between the so-called neutral theories which assume functional equivalence among coexisting species to forest simulators that explain species assemblages as the result of tradeoffs in species individual strategies at several ontogenetic stages. Progress in these questions has been hindered by the inherent difficulties of developing analytical size-structured models of stand dynamics. This precludes examination of the relative importance of each mechanism on tree coexistence. In previous simulation and analytical studies emphasis has been given to interspecific differences at the sapling stage, and less so to interspecific variation in seedling recruitment. In this study we develop a partial differential equation model of stand dynamics in which competition takes place at the recruitment stage. Species differ in their size-dependent growth rates and constant mortality rates. Recruitment is described as proportional to the basal area of conspecifics, to account for fecundity and seed supply per unit of basal area, and is corrected with a decreasing function of species specific basal area to account for competition. We first analyze conditions for population persistence in monospecific stands and second we investigate conditions of coexistence for two species. In the monospecific case we found a stationary stand structure based on an inequality between mortality rate and seed supply. In turn, intra-specific competition does not play any role on the asymptotic extinction or population persistence. In the two-species case we found that coexistence can be attained when the reciprocal negative effect on recruitment follows a given relation with respect to intraspecific competition. Specifically a tradeoff between recruitment potential (i.e. shade tolerance or predation avoidance) and fecundity or growth rate. This is to our knowledge the first study that describes coexistence mechanisms in an analytical size-structured model in terms of competitive differences at the regeneration state.

Feeding anatomy, filter-feeding rate, and diet of whale sharks Rhincodon typus during surface ram filter feeding off the Yucatan Peninsula, Mexico.

The feeding anatomy, behavior and diet of the whale shark Rhincodon typus were studied off Cabo Catoche, Yucatan Peninsula, Mexico. The filtering apparatus is composed of 20 unique filtering pads that completely occlude the pharyngeal cavity. A reticulated mesh lies on the proximal surface of the pads, with openings averaging 1.2mm in diameter. Superficial to this, a series of primary and secondary cartilaginous vanes support the pads and direct the water across the primary gill filaments. During surface ram filter feeding, sharks swam at an average velocity of 1.1m/s with 85% of the open mouth below the water's surface. Sharks on average spent approximately 7.5h/day feeding at the surface on dense plankton dominated by sergestids, calanoid copepods, chaetognaths and fish larvae. Based on calculated flow speed and underwater mouth area, it was estimated that a whale shark of 443 cm total length (TL) filters 326 m(3)/h, and a 622 cm TL shark 614 m(3)/h. With an average plankton biomass of 4.5 g/m(3) at the feeding site, the two sizes of sharks on average would ingest 1467 and 2763 g of plankton per hour, and their daily ration would be approximately 14,931 and 28,121 kJ, respectively. These values are consistent with independently derived feeding rations of captive, growing whale sharks in an aquarium. A feeding mechanism utilizing cross-flow filtration of plankton is described, allowing the sharks to ingest plankton that is smaller than the mesh while reducing clogging of the filtering apparatus.

Competition and species coexistence in a metapopulation model: can fast asymmetric migration reverse the outcome of competition in a homogeneous environment?

We investigate whether asymmetric fast migration can modify the predictions of classical competition theory and, in particular revert species dominance. We consider a model of two species competing for an implicit resource on a habitat divided into two patches. Both patches are connected through constant migration rates and in each patch local dynamics are driven by a Lotka-Volterra competition system. Local competition is asymmetric with the same superior competitor in both patches. Migration is asymmetric, species dependent and fast in comparison to local competitive interactions. The species and patches are taken to be otherwise similar: in both patches we assume the same carrying capacities for both species, and the same growth rates and pair-wise competition coefficients for each species. We show that global dynamics can be described by a classical Lotka-Volterra competition model. We found that by modifying the ratio of intraspecific migration rates for both species all possible combinations of global species relative dominance can be achieved. We find specific conditions for which the local superior competitor is globally excluded. This is to our knowledge the first study showing that fast asymmetric migration can lead to inferior competitor dominance in a homogeneous environment. We conclude that disparity of temporal scales between migration and local dynamics may have important consequences for the maintenance of biodiversity in spatially structured populations.

Effects of density-dependent migrations on stability of a two-patch predator-prey model.

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.

Approximate reduction of multiregional models with environmental stochasticity.

In this work we extend previous results regarding the use of approximate aggregation techniques to simplify the study of discrete time models for populations that live in an environment that changes randomly with time. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. We present the reduction of a stochastic multiregional model in which the population, structured by age and spatial location, lives in a random environment and in which migration is fast with respect to demography. However, the technique works in much more general settings as, for example, those of stage-structured populations living in a multipatch environment. By manipulating the original system and appropriately defining the global variables we obtain a simpler system. The paper concentrates on establishing relationships between the original and the reduced systems for a given separation of time scales between the two processes. In particular, we relate the original state variables and the global variables and, in the case the pattern of temporal variation is Markovian, we relate the presence of strong stochastic ergodicity for the original and reduced systems. Moreover, we relate different measures of asymptotic population growth for these systems.

An analytical model of stand dynamics as a function of tree growth, mortality and recruitment: the shade tolerance-stand structure hypothesis revisited.

Light competition and interspecific differences in shade tolerance are considered key determinants of forest stand structure and dynamics. Specifically two main stand diameter distribution types as a function of shade tolerance have been proposed based on empirical observations. All-aged stands of shade tolerant species tend to have steeply descending, monotonic diameter distributions (inverse J-shaped curves). Shade intolerant species in contrast typically exhibit normal (unimodal) tree diameter distributions due to high mortality rates of smaller suppressed trees. In this study we explore the generality of this hypothesis which implies a causal relationship between light competition or shade tolerance and stand structure. For this purpose we formulate a partial differential equation system of stand dynamics as a function of individual tree growth, recruitment and mortality which allows us to explore possible individual-based mechanisms--e.g. light competition-underlying observed patterns of stand structure--e.g. unimodal or inverse J-shaped equilibrium diameter curves. We find that contrary to expectations interspecific differences in growth patterns can result alone in any of the two diameter distributions types observed in the field. In particular, slow growing species can present unimodal equilibrium curves even in the absence of light competition. Moreover, light competition and shade intolerance evaluated both at the tree growth and mortality stages did not have a significant impact on stand structure that tended to converge systematically towards an inverse J-shaped curves for most tree growth scenarios. Realistic transient stand dynamics for even aged stands of shade intolerant species (unimodal curves) were only obtained when recruitment was completely suppressed, providing further evidence on the critical role played by juvenile stages of tree development (e.g. the sampling stage) on final forest structure and composition. The results also point out the relevance of partial differential equations systems as a tool for exploring the individual-level mechanisms underpinning forest structure, particularly in relation to more complex forest simulation models that are more difficult to analyze and to interpret from a biological point of view.

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics.

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.

Migration frequency and the persistence of host-parasitoid interactions.

This paper analyses the effect of migration frequency on the stability and persistence of a host-parasitoid system in a two-patch environment. The hosts and parasitoids are allowed to move from one patch to the other a certain number of times within a generation. When this number is low, i.e. when the time-scales associated with migration and demography are of the same order, host-parasitoid interactions are usually not persistent. When this number is high, however, persistence is more likely. Moreover, in this situation, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. Analysis of the aggregated model shows that the system reaches a stable steady state for some regions of the parameter domain. Persistence occurs when the movement of the parasitoids is asymmetrical, i.e. they move preferentially to one of the two patches. We show that the growth rate of the host population is a key parameter in determining which migration strategies of the parasitoids lead to persistent host-parasitoid interactions.

Linear discrete population models with two time scales in fast changing environments II: non-autonomous case.

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system. In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationships between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system.

A predator-prey model with predators using hawk and dove tactics.

In this work we present a predator-prey model that incorporates individual behavior of the predators. A classical Lotka-Volterra model with self-limiting prey describes the predator-prey interaction. Predator individuals can use two behavioral tactics to dispute a prey when they meet, the classical hawk and dove tactics. Each individual can use both tactics along its life. The predator behavioral change is described by means of a game dynamic model based upon the replicator equations, where the gain depends on prey density. We assume that the demographic process, predator-prey interactions, acts at a slow time scale in comparison with the evolution of the behavior of the predator population. The existence of two time scales allows studying the complete system from a reduced one, which describes the dynamics of the total predator and prey densities at the slow time scale. The aim of this work is to study the effects of individual predator behavior on the dynamics of the predator-prey system. The main conclusion that emerges from this study is the existence of a relationship between prey density and the strategy adopted by predators: aggressive behavior is connected to high prey and low predator densities, whereas a polymorphism dove-hawk is found at low prey and high predator densities.

The reliability of approximate reduction techniques in population models with two time scales.

As a result of the complexity inherent in some natural systems, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often find multiregional models for structured populations in which individuals are classified regarding their age and their spatial location. Dealing with such structured populations leads to high dimensional models. Moreover, in many instances the dynamics of the system is controlled by processes whose time scales are very different from each other. For example, in multiregional models migration is often a fast process in comparison to the growth of the population. Approximate reduction techniques take advantage of the presence of different time scales in a system to introduce approximations that allow one to transform the original system into a simpler low dimensional system. In this way, the dynamics of the original system can be approximated in terms of that of the reduced system. This work deals with the study of that approximation. In particular, we work with a non-autonomous discrete time model previously presented in the literature and obtain different bounds for the error we incur when we describe the dynamics of the original system in terms of the reduced one. The results are illustrated by some numerical simulations corresponding to the reduction of a Leslie type model for a population structured in two age classes and living in a two patch system.