Kei-ichi Tainaka

Indirect relation between species extinction and habitat destruction

To study local destruction of habitat, we present a lattice ecosystem composed of prey (X) and predator (Y). This system corresponds to a lattice version of the Lotka–Volterra model, where interaction is allowed between neighboring lattice points. The lattice is partly destroyed, and destructed sites or barriers are randomly located between adjacent lattice points with the probability p. The barrier interrupts the reproduction of X, but the species Y suffers no direct damage by barriers. This system exhibits an extinction due to an indirect effect: when the density p of barriers increases, the species Y goes extinct. On the other hand, an initial suppression of X may later lead to the increase of X. The predator Y decreases in spite of the increase of X. These results cannot be explained by a mean-field theory such as the Lotka–Volterra equation. We discuss that endangered species may become extinct by a slight perturbation to their habitat.

Spatial coexistence of phytoplankton species in ecological timescale

The species diversity of phytoplankton is usually very high in wild aquatic systems, as seen in the paradox of plankton. Coexistence of many competitive phytoplankton species is extremely common in nature. However, experiments and mathematical theories show that interspecific competition often leads to the extinction of most inferior species. Here, we present a lattice version of a multi-species Lotka–Volterra competition model to demonstrate the importance of local interaction. Its mathematical equilibrium is the exclusion of all but one superior species. However, temporal coexistence of many competitive species is possible in an ecological time scale if interactions are local instead of global. This implies that the time scale is elongated many orders when interactions are local. Extremely high species diversity of phytoplankton in aquatic systems may be maintained by spatial coexistence in an ecological time scale.

Allee effect in the selection for prime-numbered cycles in periodical cicadas

Periodical cicadas are well known for their prime-numbered life cycles (17 and 13 years) and their mass periodical emergences. The origination and persistence of prime-numbered cycles are explained by the hybridization hypothesis on the basis of their lower likelihood of hybridization with other cycles. Recently, we showed by using an integer-based numerical model that prime-numbered cycles are indeed selected for among 10- to 20-year cycles. Here, we develop a real-number-based model to investigate the factors affecting the selection of prime-numbered cycles. We include an Allee effect in our model, such that a critical population size is set as an extinction threshold. We compare the real-number models with and without the Allee effect. The results show that in the presence of an Allee effect, prime-numbered life cycles are most likely to persist and to be selected under a wide range of extinction thresholds.

Selection for prime-number intervals in a numerical model of periodical cicada evolution.

Periodical cicadas are known for unusually long and prime-numbered life cycles (13 and 17 years) for insects. To explain the evolution of prime-numbered reproductive intervals (life cycles), the hybridization hypothesis claims that prime numbers greatly reduce the chance of hybridization with other life cycles. We investigate the hybridization hypothesis using a simulation model. This model is a deterministic, discrete population model with three parameters: larval survival per year, clutch size, and emergence success. Reproductive intervals from 10 years to 20 years compete for survival in the simulations. The model makes three key assumptions: a Mendelian genetic system, random mating among broods of different life-cycle lengths, and integer population sizes. Longer life cycles have larger clutch sizes but suffer higher total mortality than shorter life cycles. Our results show that (1) nonprime-numbered reproductive intervals disappear rapidly in comparison to the selection among the various prime-numbered life cycles, (2) the selection of prime-numbered intervals happens only when populations are at the verge of extinction, and (3) the 13- and 17-year prime phenotypes evolve under certain conditions of the model and may coexist. The hybridization hypothesis is discussed in light of other hypotheses for the evolution of periodical cicada life cycles.

Patch dynamics based on Prisoner's Dilemma game: superiority of golden rule

There has been much literature on ecological model of Prisoners Dilemma (PD) game. This game illustrates that cooperation can evolve in situations where individuals tend to look after themselves. In order to explain some behaviors of altruism in animal societies, the strategy All Cooperate (AC), often called the Golden Rule, is more appropriate than other strategies. However, very little is known about the superiority of AC. In the present article, we study patch dynamics based on non-iterated PD game, applying two different methods: island and lattice models. Each patch is assumed to be either vacant or composed of a population of AC or All Defect (AD), where AD means a selfish strategy. Both models exhibit a phase transition between a phase where both AC and AD survive, and a phase where AD is extinct. The latter phase means that AC beats AD completely. In the case of lattice model, the extinction of AD easily occurs and the abundance of AC takes a larger value, compared with the island model. Our models can be also extended to general iterated PD game; we describe the reason why AC can outperform any other strategy.

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.

Physics and Ecology of Rock-Paper-Scissors Game

From physical and ecological aspects, we reviewan interacting particle system which follows a rule of the Rock-Paper-Scissors (RPS) game. This rule symbolically represents a food chain in ecosystems. It also represents nonequilibrium systems which have a feedback mechanism.We describe the spatial pattern dynamics in lattice RPS system: the time dependence of each species is not fully understood, especially on two-dimensional lattice. Moreover, we modify and apply RPS rule to voter and biological systems. Computer simulation for both voter model and ecosystems exhibits counter-intuitive results in phase transition. Such results can be seen in many cyclic systems, and they may be related to the unpredictability in nonequilibrium systems.

Segregation in an interacting particle system

Abstract We present a lattice system composed of three kinds of particles which interact following a rule combining the contact process with Rock–Paper–Scissors game. Depending on value of a parameter, this system naturally evolves into a specific pattern, where segregation of three species occurs. We discuss that such a segregation phenomenon associates with a habitat isolation of biospecies.

Dynamic process and variation in the contact process

Abstract Dynamic process in one-dimensional contact process (CP) is studied by computer simulation. We carry out the experiment of phase transition by two different methods: CP and its mean-field version. It is known that for both methods, stationary state exhibits the critical slowing-down; relaxation time diverges near the phase boundary. In the present article, we find only for CP that the enhancement of fluctuation occurs in the transient state of dynamical process.

Breeding games and dimorphism in male salmon

In certain species of salmon, male phenotypes occur in two distinct morphs: the large ‘hooknose’ or the small ‘jack’. Hooknoses fight each other for access to females, while jacks occupy refuges near spawning beds to sneak fertilizations. Jacks also fight each other over opportunities for sneaking without immediate gains. To explore whether the jack behavioural strategy is equally adaptive to that of the hooknose, we built a game-theoretic model similar to the classic hawk–dove game, with and without conditions of density dependence. Our model demonstrates that fitness of the jack strategy increases with the frequency of the hooknose strategy, because jacks can steal the benefits otherwise accrued by hooknoses. The coexistence of strategies is much more easily achieved in this game than in the hawk–dove game. When negative density effects on benefits are introduced to the model, coexistence conditions are further relaxed. Hence, the jack and hooknose strategies can be viewed as equally adaptive, resulting in a stable mixed evolutionarily stable strategy (ESS).