PHYSICAL REVIEW E 97, 052413 (2018) General two-species interacting Lotka-Volterra system: Population dynamics and wave propagation Haoqi Zhu, 1 Mao-Xiang Wang, 1 , * and Pik-Yin Lai 2 , † 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 Department of Physics and Center for Complex Systems, National Central University, Chung-Li District, Taoyuan City 320, Taiwan, Republic of China (Received 2 February 2018; revised manuscript received 8 May 2018; published 25 May 2018) The population dynamics of two interacting species modeled by the Lotka-Volterra (LV) model with general parameters that can promote or suppress the other species is studied. It is found that the properties of the two species’ isoclines determine the interaction of species, leading to six regimes in the phase diagram of interspecies interaction; i.e., there are six different interspecific relationships described by the LV model. Four regimes allow for nontrivial species coexistence, among which it is found that three of them are stable, namely, weak competition, mutualism, and predator-prey scenarios can lead to win-win coexistence situations. The Lyapunov function for general nontrivial two-species coexistence is also constructed. Furthermore, in the presence of spatial diffusion of the species, the dynamics can lead to steady wavefront propagation and can alter the population map. Propagating wavefront solutions in one dimension are investigated analytically and by numerical solutions. The steady wavefront speeds are obtained analytically via nonlinear dynamics analysis and verified by numerical solutions. In addition to the inter- and intraspecific interaction parameters, the intrinsic speed parameters of each species play a decisive role in species populations and wave properties. In some regimes, both species can copropagate with the same wave speeds in a finite range of parameters. Our results are further discussed in the light of possible biological relevance and ecological implications. DOI: 10.1103/PhysRevE.97.052413 I. INTRODUCTION The population of a species is often affected by other species forming an ecological web in nature. Furthermore, the population dynamics of a certain species is strongly dependent on another species that is directly interacting. Such a two-species interacting system has received extensive interest both in theories and in ecological observations. In particular, the Lotka-Volterra (LV) model [1–3] is believed to be an appropriate model to study such an interacting community [4,5]. In addition to population dynamics, LV models have been employed in different contexts, such as evolutionary game theory [6–8], food webs [9], and replicator equations [10]. While the dynamics and stability of the LV system for the case of well-mixed populations have been rather well investigated in some situations, such as competitive, cooperative, and predator-prey interactions by various groups, there is little or no complete study on the system with different possible interaction parameters under a unified description. In addition, the case in which the species can undergo spatial diffusion are much less studied. Motility or diffusive motion can drastically alter the temporal dynamics and spatial patterns, and in some situations traveling wave or directed motion can propagate [11,12]. The spatiotemporal patterns resulting from the inter- and intrainteraction dynamics of two species (or agents) are also of recent interest in other dynamical systems, such as evolutionary game theory [8], replicator dynamics [13], and in * wangmx@njust.edu.cn † pylai@phy.ncu.edu.tw general reaction diffusion systems. For example, spatial diffu- sion can lead to various spatiotemporal pattern formations [14], spiral waves in complex Ginzburg-Landau equations [15], and cooperation patterns in prisoners’ dilemma dynamics [16,17]. It is known that for LV model with diffusion, there exists traveling wave solutions propagating from one stationary point to another [18–25]. At first glance, diffusion is a kind of random motion that should not be associated with directed motion. However, nonlinearity resulted from the interactions between the species can produce propagating waves, which travel much faster than the species’ diffusional speeds. Such a propagating wavefront represents a progressive replacement of one equilibrium (ahead of the front) by another (behind the front). Moreover, it has been shown that propagating wavefront can exist for the competitve [24] and mutualistic [25] LV system with diffusion. It would be of interest to find out other possible wave dynamics and their properties for species interactions in addition to the competitive and mutualistic ones. In this paper, we consider the LV model of two interacting species with spatial diffusion and investigate the population dynamics and steady wavefront propagation. One of the aims is to summarize all the interaction types between two species described by the LV model and obtain the phase diagram to provide deeper insights in the interacting mechanism in a two- species community. Another goal is to figure out the conditions and properties of wave propagation in the presence of species diffusion for general species interactions. In particular, we shall classify different dynamical scenarios under a unified phase diagram and investigate the wavefront profiles and wave speeds. Analytical results for the wavefront and waveback speeds are obtained using nonlinear dynamics techniques and 2470-0045/2018/97(5)/052413(15) 052413-1 ©2018 American Physical Society