arXiv:2212.07630v1 [math.DS] 15 Dec 2022 Competitive Coexistence In Lotka-Volterra Model Incorporating Type II functional Response Gholam Reza Rokni Lamouki * Mahmoud Soufbaf † Khosro Tajbakhsh ‡ December 16, 2022 Abstract A competitive resource-consumer dynamical model is analyzed based on a novel Lotka-Volterra model similar to Rosenwig-McArthur one. Resource logistic growth in the absence of consumers is logical and widely used by re- searchers. However, type II Holling’s functional response to competing consumers made the model structure more realistic. We used the normal-form and the center manifold theorems for bifurcation analysis of the presented model, identified Hopf and zero-Hopf bifurcations and their directions, and discussed their biological interpretations. We hy- pothesized that differentiated time scales of the competing consumers’ predatory are the mechanisms that promote co- existence through relaxation-oscillation dynamics. We showed that with very asymmetric competition (highly different coefficient α and β values), in all cases and under symmetric attack rates, the competitive exclusion did not happen in intervals between each relaxation oscillation. Graphical representation of variations of the first Lyapunov coefficient, af- ter competition coefficients interplay, shows various dynamics with growing complexity from the periodic state towards chaotic motion like R¨ ossler attractor. In such cases, we found that Smale claim on the competetive systems could be true even for three dimensional systems with two competitors in which functional response is nonlinear. We presented simulations to visualize the theoretical results obtained through bifurcation analysis. Keywords:Competition, First Lyapunov Coefficient, Functional Response, Hopf Bifurcation, Zero-Hopf, Relaxation- Oscillation 1 Introduction Competition among organisms for limited common resources has been a controversial issue among researchers for decades through many hypotheses and tests. Generally, competition between species takes place through two main paths. The first is the exploitative/exploratory competition which involves indirect negative interactions. Here each consumer affects the other by reducing the abundance of the resource. The second is interference, which involves direct negative interactions between species due to mechanisms such as territoriality, predation, and chemical competition; see [1], [2], and [3]. There are few theories of interference competition that practically follow the Lotka-Volterra tradition. They do not consider the resource accumulation within the system where the resource is an input rather than a status variable [1]. The classical competition theory predicts that competing species can only coexist through the mechanisms such as re- source allocation in time and space, differences in the emergence and attack rates, as well as distribution and search efficiency [4][3]. However, the microbial competition models studied in a chemostat show that competition eliminates the population that needs a higher concentration of nutrients to grow [9]. Biological reasons such as complex interac- tions between trophic levels in the food webs disrecommend the study based on a single phenomenon [39]. This idea extends to the competition phenomenon. It is common to believe that the diversity of food resources can explain the high biodiversity in animals [21]; however, the enriched biodiversity of plant species that all require the same physical and chemical resources is unclear [22]. Researchers believe that the persistent disruption has created mosaics of localized re- sources. Then, due to spatial arrangement and partitioning of competitors on respective accessible resources, competitive exclusion avoids occur [24]. The importance of spatial processes in population dynamics is a concern to ecologists. It is now well established that the qualitative behavior of interactions between species such as host-parasite, prey-predator, and competition in a heterogeneous environment is quite different from the corresponding behavior in a homogeneous one [24]. For instance, [29] suggested that parasitoids may be able to coexist on the same host species if they partition host resources according to size, age, and stage; or if their dynamics vary at spatial and temporal scales. However, re- ducing niche overlap mainly by weaker competitors is suggested as a priory mechanism promoting the Spatio-temporal coexistence of both species in a community [31]. There are many ways to incorporate non-uniformity into models of the consumer population. The time delay in the consumer response to the change in resource density and the relatively slow * School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, rokni@ut.ac.ir † Agriculture Research School, Nuclear Science and Technology Research Institute, Karaj, Iran, msoufbaf@aeoi.org.ir ‡ Faculty of Mathematical Sciences, Tarbiat Modares University, 14115-134 Tehran, Iran,khtajbakhsh@tmu.ac.ir 1