Population persistence under two conservation measures: Paradox of habitat protection in a patchy environment.

Anthropogenic modification of natural habitats is a growing threat to biodiversity and ecosystem services. The protection of biospecies has become increasingly important. Here, we pay attention to a single species as a conservation target. The species has three processes: reproduction, death and movement. Two different measures of habitat protection are introduced. One is partial protection in a single habitat (patch); the mortality rate of the species is reduced inside a rectangular area. The other is patch protection in a two-patch system, where only the mortality rate in a particular patch is reduced. For the one-patch system, we carry out computer simulations of a stochastic cellular automaton for a "contact process". Individual movements follow random walking. For the two-patch system, we assume an individual migrates into the empty cell in the destination patch. The reaction-diffusion equation (RDE) is derived, whereby the recently developed "swapping migration" is used. It is found that both measures are mostly effective for population persistence. However, comparing the results of the two measures revealed different behaviors. ⅰ) In the case of the one-patch system, the steady-state densities in protected areas are always higher than those in wild areas. However, in the two-patch system, we have found a paradox: the densities in protected areas can be lower than those in wild areas. ⅱ) In the two-patch system, we have found another paradox: the total density in both patches can be lower, even though the proportion of the protected area is larger. Both paradoxes clearly occur for the RDE with swapping migration.

Serious role of non-quarantined COVID-19 patients for random walk simulations.

The infectious disease (COVID-19) causes serious damages and outbreaks. A large number of infected people have been reported in the world. However, such a number only represents those who have been tested; e.g. PCR test. We focus on the infected individuals who are not checked by inspections. The susceptible-infected-recovered (SIR) model is modified: infected people are divided into quarantined (Q) and non-quarantined (N) agents. Since N-agents behave like uninfected people, they can move around in a stochastic simulation. Both theory of well-mixed population and simulation of random-walk reveal that the total population size of Q-agents decrease in spite of increasing the number of tests. Such a paradox appears, when the ratio of Q exceeds a critical value. Random-walk simulations indicate that the infection hardly spreads, if the movement of all people is prohibited ("lockdown"). In this case the infected people are clustered and locally distributed within narrow spots. The similar result can be obtained, even when only non-infected people move around. However, when both N-agents and uninfected people move around, the infection spreads everywhere. Hence, it may be important to promote the inspections even for asymptomatic people, because most of N-agents are mild or asymptomatic.

Microbial mutualism promoting the coexistence of competing species: Double-layer model for two competing hosts and one microbial species.

Gause's law of competitive exclusion holds that the coexistence of competing species is extremely unlikely when niches are not differentiated. This law is supported by many mathematical studies, yet the coexistence of competing species is nearly ubiquitous in real ecosystems. We pay attention to the fact that plants and animals usually contact with microbial species as mutualistic partners. The activity spaces of host species are different from those of micro-organisms. In the present study, we apply double-layer model to the association of two competing hosts and a microorganism. Two lattices are prepared: one is for hosts, and the other is for microorganism. The basic equation obtained by mean-field theory is an extension of Lotka-Volterra competition model. Both mathematical analysis and numerical simulations reveal that a shared microbial mutualist can permit the coexistence of competing hosts. From the derived condition of coexistence, we believe the microbial mutualism promotes biodiversity in many ecological systems.

Effects of pest control on a food chain in patchy environment: Species-dependent activity range on multilayer graphs.

Ecosystems on earth are strongly affected by human life. We pay attention to pest control in a patchy environment. To date, many authors have reported the indeterminacy in pest control. Most of these works have been studied in single-habitat systems. In the present article, however, we consider a food chain model (prey, predator and top predator) on five networks of patches, where node and link denote habitable patch and migration path, respectively. Each network includes three layers which represent the activity ranges of respective species. Reaction-migration equations are solved analytically and numerically. It is found the dynamics largely change depending on the geometry of networks. When removal rate of top predator is increased, the so-called "top-down effect" is commonly observed. In this case, the pest control will be successful, but extinction point of top predator largely differs on different networks. When removal rate of intermediate predator is increased, the responses of system become complicated. The responses differ not only for each patch but also for each geometry. Hence, the pest control on intermediate predators may fail.

Metapopulation model for a prey-predator system: Nonlinear migration due to the finite capacities of patches.

Many species live in spatially separated patches, and individuals can migrate between patches through paths. In real ecosystems, the capacities of patches are finite. If a patch is already occupied by the individuals of some species, then the migration into the patch is impossible. In the present paper, we deal with prey-predator system composed of two patches. Each patch contains a limited number of cells, where the cell is either empty or occupied by an individual of prey or predator. We introduce "swapping migration" defined by the exchange between occupied and empty cells. An individual can migrate, only when there are empty cells in the destination patch. Reaction-migration equations in prey-predator system are presented, where the migration term forms nonlinear function of densities. We numerically solve equilibrium densities, and find that the population dynamics are largely affected by nonlinear migration. Not only extinction points but also the responses to the environmental changes crucially depend on the patch capacities.

Metapopulation dynamics in the rock-paper-scissors game with mutation: Effects of time-varying migration paths.

Migration paths of animals are rarely the same. The paths may change according to seasonal and circadian rhythms. We study the effect of temporal migration on population dynamics of rock-paper-scissors (RPS) games with mutation by using the metapopulation dynamic model with two patches. Via mutation, an individual R changes to S with rate µ. All agents move by random walk between two patches and the RPS game is performed in each patch. The migration path between two patches is switched on or off periodically. The dynamics are represented by the reaction-diffusion equations with time-dependent diffusion coefficients in diffusively coupled reactors. We obtain the solutions of time-dependent reaction-diffusion equations numerically and analytically. The time-varying migration path induces complex behavior for the RPS dynamics, depending on the frequency of the periodical path. We find that the phase transitions occur by varying mutation rate µ. The phase transition depends highly on the frequency.

Metapopulation model of rock-scissors-paper game with subpopulation-specific victory rates stabilized by heterogeneity.

Recently, metapopulation models for rock-paper-scissors games have been presented. Each subpopulation is represented by a node on a graph. An individual is either rock (R), scissors (S) or paper (P); it randomly migrates among subpopulations. In the present paper, we assume victory rates differ in different subpopulations. To investigate the dynamic state of each subpopulation (node), we numerically obtain the solutions of reaction-diffusion equations on the graphs with two and three nodes. In the case of homogeneous victory rates, we find each subpopulation has a periodic solution with neutral stability. However, when victory rates between subpopulations are heterogeneous, the solution approaches stable focuses. The heterogeneity of victory rates promotes the coexistence of species.

Heterogeneous network promotes species coexistence: metapopulation model for rock-paper-scissors game.

Understanding mechanisms of biodiversity has been a central question in ecology. The coexistence of three species in rock-paper-scissors (RPS) systems are discussed by many authors; however, the relation between coexistence and network structure is rarely discussed. Here we present a metapopulation model for RPS game. The total population is assumed to consist of three subpopulations (nodes). Each individual migrates by random walk; the destination of migration is randomly determined. From reaction-migration equations, we obtain the population dynamics. It is found that the dynamic highly depends on network structures. When a network is homogeneous, the dynamics are neutrally stable: each node has a periodic solution, and the oscillations synchronize in all nodes. However, when a network is heterogeneous, the dynamics approach stable focus and all nodes reach equilibriums with different densities. Hence, the heterogeneity of the network promotes biodiversity.

Asymptotic stability of a modified Lotka-Volterra model with small immigrations.

Predator-prey systems have been studied intensively for over a hundred years. These studies have demonstrated that the dynamics of Lotka-Volterra (LV) systems are not stable, that is, exhibiting either cyclic oscillation or divergent extinction of one species. Stochastic versions of the deterministic cyclic oscillations also exhibit divergent extinction. Thus, we have no solution for asymptotic stability in predator-prey systems, unlike most natural predator-prey interactions that sometimes exhibit stable and persistent coexistence. Here, we demonstrate that adding a small immigration into the prey or predator population can stabilize the LV system. Although LV systems have been studied intensively, there is no study on the non-linear modifications that we have tested. We also checked the effect of the inclusion of non-linear interaction term to the stability of the LV system. Our results show that small immigrations invoke stable convergence in the LV system with three types of functional responses. This means that natural predator-prey populations can be stabilized by a small number of sporadic immigrants.

Epidemics of random walkers in metapopulation model for complete, cycle, and star graphs.

We present the metapopulation dynamic model for epidemic spreading of random walkers between subpopulations. A subpopulation is represented by a node on a graph. Each agent or individual is either susceptible (S) or infected (I). All agents move by random walk on the graph; namely, each agent randomly determines the destination of migration. The reaction-diffusion equations are presented as ordinary differential equations, not partial differential equations. To evaluate the risk of each subpopulation (node), we obtain the solutions of reaction-diffusion equations analytically and numerically for small, complete, cycle and star graphs. If a graph is homogeneous, or if every node has the same degree, then the solution never changes for any nodes. However, when a graph is heterogeneous, the infection density in equilibrium differs entirely among nodes. For example, on star graphs, the hub seems to be a supply source of disease because the infection density at the hub is much higher than that at the other nodes. On every graph, the epidemic thresholds are identical for all nodes.

Metapopulation model for rock-paper-scissors game: Mutation affects paradoxical impacts.

The rock-paper-scissors (RPS) game is known as one of the simplest cyclic dominance models. This game is key to understanding biodiversity. Three species, rock (R), paper (P) and scissors (S), can coexist in nature. In the present paper, we first present a metapopulation model for RPS game with mutation. Only mutation from R to S is allowed. The total population consists of spatially separated patches, and the mutation occurs in particular patches. We present reaction-diffusion equations which have two terms: reaction and migration terms. The former represents the RPS game with mutation, while the latter corresponds to random walk. The basic equations are solved analytically and numerically. It is found that the mutation induces one of three phases: the stable coexistence of three species, the stable phase of two species, and a single-species phase. The phase transitions among three phases occur by varying the mutation rate. We find the conditions for coexistence are largely changed depending on metapopulation models. We also find that the mutation induces different paradoxes in different patches.

Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects.

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.

Territory holders and non-territory holders in Ayu fish coexist only in the population growth process due to hysteresis.

Ayu fish form feeding territories during a non-breeding (growing) season. When the density of the fish increases, phases gradually change. In the early growing season, all fish can hold territories at low density. Once all territory sites are occupied, newcomers become floaters. As the density further increases, territory holders have to spend much more time in defending their own territory and lose the time to feed on algae. Eventually, all fish give up their own territories and then form a school. In contrast, when the density decreases, territories are directly reformed from the school. In short, ayu fish exhibit a different transition, called hysteresis, where the two transitions occur widely-apart from each other. The dynamics of this intrinsic phenomena has not been demonstrated in previous studies. We develop a rate equation to describe the population dynamics within territorial competition. Our model successfully reproduces territorial hysteresis and indicates that territory holders and floaters can coexist only in the process of population growth. Moreover, we also find that the two critical densities of territorial hysteresis are conspicuously different from each other when the increase of the density of floaters sharply influences (step-function-like) the territories.

Effect of directional migration on Lotka-Volterra system with desert.

Migration is observed across many species. Several authors have studied ecological migration by applying cellular automaton (CA). In this paper, we present a directional migration model with desert on a one-dimensional lattice where a traffic CA model and a lattice Lotka-Volterra system are connected. Here predators correspond to locomotive animals while prey is immobile plants. Predators migrate between deserts and fertile lands repeatedly. Computer simulations reveal the two types of phase transition: coexistence of both species and prey dominance, which is caused by both benefit and cost of migration. In the coexistence phase, the steady-state density of predators usually increases by migration as long as the desert size is small and their mortality rate is low. In contrast, the prey density increases, even if the desert size becomes large. Such a paradox comes from the indirect effect: predators go extinct by the increase of desert size, so that the plant density can increase. Moreover, we find several self-organized spatial patterns: 1) predators form a stripe pattern; namely swarms. 2) The velocity of predators is high on deserts, but very low on fertile land. 3) Predators give birth only on fertile lands.

The Effects of Rainfall on the Population Dynamics of an Endangered Aquatic Plant, Schoenoplectus gemmifer (Cyperaceae).

The conservation of aquatic plants in river ecosystems should consider the wash-out (away) problem resulting from severe rainfall. The aquatic plant Schoenoplectus gemmifer is an endangered species endemic to Japan. Our previous study reported that the population size of S. gemmifer in Hamamatsu city, Japan, had decreased by one-tenth because many individuals had been washed out by a series of heavy rains in 2004. However, there is insufficient information on the ecological nature of this endangered aquatic plant for adequate conservation. In this paper, we report the population dynamics of one population in Hamamatsu city from 2004 to 2012 in relation to rainfall. We surveyed the number and growing location of all living individuals in the population 300 times during the study period. To examine the temporal changes of individual plants, we also counted the number of culms for 38 individuals in four observations among 300 records. Decreases and increases in the population size of this plant were associated with washing out and the settlement of gemmae (vegetative propagation), respectively. The major cause of the reduction in the population size was an increase in the number of washed-out individuals and not the decreased settlement of gemmae. The wash-out rates for small and large individuals were not significantly different. Small individuals having a stream form with linear leaves resisted flooding, and large individuals were often partially torn off by flooding events. Modification of river basins to reduce the flow velocity may be effective for the conservation of S. gemmifer.

The paradox of enrichment in phytoplankton by induced competitive interactions.

The biodiversity loss of phytoplankton with eutrophication has been reported in many aquatic ecosystems, e.g., water pollution and red tides. This phenomenon seems similar, but different from the paradox of enrichment via trophic interactions, e.g., predator-prey systems. We here propose the paradox of enrichment by induced competitive interactions using multiple contact process (a lattice Lotka-Volterra competition model). Simulation results demonstrate how eutrophication invokes more competitions in a competitive ecosystem resulting in the loss of phytoplankton diversity in ecological time. The paradox is enhanced under local interactions, indicating that the limited dispersal of phytoplankton reduces interspecific competition greatly. Thus, the paradox of enrichment appears when eutrophication destroys an ecosystem either by elevated interspecific competition within a trophic level and/or destabilization by trophic interactions. Unless eutrophication due to human activities is ceased, the world's aquatic ecosystems will be at risk.

Evolutionary trajectories explain the diversified evolution of isogamy and anisogamy in marine green algae.

The evolution of anisogamy (the production of gametes of different size) is the first step in the establishment of sexual dimorphism, and it is a fundamental phenomenon underlying sexual selection. It is believed that anisogamy originated from isogamy (production of gametes of equal size), which is considered by most theorists to be the ancestral condition. Although nearly all plant and animal species are anisogamous, extant species of marine green algae exhibit a diversity of mating systems including both isogamy and anisogamy. Isogamy in marine green algae is of two forms: isogamy with extremely small gametes and isogamy with larger gametes. Based on disruptive selection for fertilization success and zygote survival (theory of Parker, Baker, and Smith), we explored how environmental changes can contribute to the evolution of such complex mating systems by analyzing the stochastic process in the invasion simulations of populations of differing gamete sizes. We find that both forms of isogamy can evolve from other isogamous ancestors through anisogamy. The resulting dimensionless analysis accounts for the evolutionary stability of all types of mating systems in marine green algae, even in the same environment. These results imply that evolutionary trajectories as well as the optimality of gametes/zygotes played an important role in the evolution of gamete size.

Apoptosis at inflection point in liquid culture of budding yeasts.

Budding yeasts are highly suitable for aging studies, because the number of bud scars (stage) proportionally correlates with age. Its maximum stages are known to reach at 20-30 stages on an isolated agar medium. However, their stage dynamics in a liquid culture is virtually unknown. We investigate the population dynamics by counting scars in each cell. Here one cell division produces one new cell and one bud scar. This simple rule leads to a conservation law: "The total number of bud scars is equal to the total number of cells." We find a large discrepancy: extremely fewer cells with over 5 scars than expected. Almost all cells with 6 or more scars disappear within a short period of time in the late log phase (corresponds to the inflection point). This discrepancy is confirmed directly by the microscopic observations of broken cells. This finding implies apoptosis in older cells (6 scars or more).

Life cycle replacement by gene introduction under an allee effect in periodical cicadas.

Periodical cicadas (Magicicada spp.) in the USA are divided into three species groups (-decim, -cassini, -decula) of similar but distinct morphology and behavior. Each group contains at least one species with a 17-year life cycle and one with a 13-year cycle; each species is most closely related to one with the other cycle. One explanation for the apparent polyphyly of 13- and 17-year life cycles is that populations switch between the two cycles. Using a numerical model, we test the general feasibility of life cycle switching by the introduction of alleles for one cycle into populations of the other cycle. Our results suggest that fitness reductions at low population densities of mating individuals (the Allee effect) could play a role in life cycle switching. In our model, if the 13-year cycle is genetically dominant, a 17-year cycle population will switch to a 13-year cycle given the introduction of a few 13-year cycle alleles under a moderate Allee effect. We also show that under a weak Allee effect, different year-classes ("broods") with 17-year life cycles can be generated. Remarkably, the outcomes of our models depend only on the dominance relationships of the cycle alleles, irrespective of any fitness advantages.

Historical effect in the territoriality of ayu fish.

Ayu fish form algae-feeding territories in a river during a non-breeding (growing) season. We build a cost-benefit theory to describe the breakdown and formation of territory. In the early stage of a growing season, all fish hold territories at low densities. Once all territory sites are occupied, excess fish become floaters. When fish density further increases, a phase transition occurs: all the territories suddenly break down and fish form a school. In contrast, when the fish density is decreased, territories are suddenly formed from the school. Both theory and experiments demonstrate that ayu should exhibit a historical effect: the breakdown and formation processes of territory are largely different. In particular, the theory in formation process predicts a specific fish behavior: an "attempted territory holder" that tries to have a small territory emerges just before the formation of territory.