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The Influence of Density in Population Dynamics with Strong and Weak Allee Effect

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The Influence of Density in Population Dynamics with Strong and Weak Allee Effect Keya et al. J Egypt Math Soc (2021) 29:4 https://doi.org/10.1186/s42787-021-00114-x Journal of the Egyptian Mathematical Society ORIGINAL RESEARCH Open Access The infuence of density in population dynamics with strong and weak Allee efect Kamrun Nahar Keya1, Md. Kamrujjaman2* and Md. Shafqul Islam3 *Correspondence: [email protected] Abstract 2 Department In this paper, we consider a reaction–difusion model in population dynamics and of Mathematics, University of Dhaka, Dhaka 1000, study the impact of diferent types of Allee efects with logistic growth in the hetero- Bangladesh geneous closed region. For strong Allee efects, usually, species unconditionally die out Full list of author information and an extinction-survival situation occurs when the efect is weak according to the is available at the end of the article resource and sparse functions. In particular, we study the impact of the multiplicative Allee efect in classical difusion when the sparsity is either positive or negative. Nega- tive sparsity implies a weak Allee efect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee efect, and the popula- tion extinct without any condition. The infuence of Allee efects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability condi- tions and the region of positive solutions (multiple solutions may exist) are presented. When the difusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results. Keywords: Strong and weak Allee efect, Regular difusion, Global analysis, Bifurcation analysis, Extinction, Survival AMS Subject Classifcation: 92D25, 35K57 (primary), 37N25 Introduction In population biology, a small or sparse population for some species is difcult to study because many factors such as mating may be difcult to analyze. A small or sparse popu- lation is one most important reasons for the extinction of the population and is known as the Allee efect. Allee efects are broadly defned as a decline in individual ftness at low population size or low density, which can result in critical population thresholds below which populations crash to extinction. Tere are many reasons by which an Allee efect can arise including • an Allee efect begins with reproductive mechanisms, including fertilization ef- ciency in broadcast spawners. For example, eggs of aquatic animals (such as fshes or oysters) that lay many small eggs; © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- ted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Keya et al. J Egypt Math Soc (2021) 29:4 Page 2 of 26 • pollen limitation and mate fnding; • sperm limitation, reproductive facilitation by conspecifcs, and female choice; • mechanisms related to survival: environmental conditioning and particularly preda- tion such as focking, coloniality, and group vigilance; • fnally, Allee efects are in social and cooperative species, where group size is impor- tant for both reproduction and survival. Te American ecologist, Warder Clyde Allee [1], was the frst researcher who presented experimental studies on Alle efects in 1931. Since then many researchers studied Allee efects in diferent frameworks. For two reasons, many of the early results on Allee efects were theoretical. First, demonstrating an Allee efect in the wild usually requires a long and complete data set of population dynamics; Second, population dynamicists were not interested in studying under-crowding, which was regarded as an interesting but anecdotal process. Te studies during the early nineties (before the real start of the upsurge), most of the researchers were concerned about marine invertebrates, plants and insects [2–4]. Only very few studied about vertebrates, except for results on fsheries [5]. During this time, a few theoretical studies were also published [6, 7]. Te Allee efect is an interesting and challenging topic in population dynamics and many researchers studied Allee efects in many diferent techniques, both theoreti- cally and experimentally. After the dominance of experimental and theoretical stud- ies on Allee efects, researchers now focusing their attention on two new directions: (i) theoretical studies via mathematical models of ever increasing details; and (ii) empiri- cal studies with the unveiling of Allee efects in natural populations. Much progress has been made by researches under these new ideas. Researchers now have a better under- standing of Allee efects and they can draw better distinctions. Note that individual ft- ness is assumed higher at low densities because of lower intra-specifc competition. Tere are several kinds of Allee efects such as component Allee efect, demographic Allee efect, dormant Allee efect, etc. and there are diferent types of relationships between these Allee efects. A dormant Allee efect is a component Allee efect that either does not result in a demographic Allee efect or results in a weak Allee efect. If interacting with a strong Allee efect, causes the overall Allee threshold to be higher than the Allee threshold due to the strong Allee efect alone. Also, a strong Allee efect is a demographic Allee efect with an Allee threshold whereas a weak Allee efect is a demo- graphic Allee efect without an Allee threshold. For details see Fig. 1. A demographic Allee efect is weak if at low density the per capita population growth rate is lower than at higher densities, but remains positive (dotted curve on Fig. 1). On the contrary, it can become so low as to become negative below a certain value, called the Allee threshold (dashed curve on Fig. 1): it is then a strong Allee efect ( [8], chap- ter 3). If a population subject to a strong Allee efect drops below that threshold, the population growth rate becomes negative and the population will get smaller at a hasten rate until it reaches zero or extinction. Very recently a group of ecologists experimented demographic and component Allee efect in group-level species and they found that at all levels, component Allee efect may generate a demographic Allee efect. A group- level demographic Allee efect always implies at least one group or subgroup-level com- ponent Allee efect [9]. Keya et al. J Egypt Math Soc (2021) 29:4 Page 3 of 26 Fig. 1 Classical negative density dependence (solid) compared to strong (dashed) and weak (dotted) Allee efects [8] Experimentally, it is easier to demonstrate that an Allee efect is strong than that an Allee efect is weak. With adequate time, we can easily trace the density for which the per capita population growth rate becomes negative. Estimates of the Allee threshold are usually approximate because low abundance will yield high observation error, fuc- tuations due to demographic stochasticity, and a signifcant proportion of counts at zero [10]. Although having an impressive dataset, the study on the island fox could not iden- tify the value of the Allee threshold even though the Allee efect was strong in some populations because these populations were already declining [11]. A strong Allee efect has also been shown in the gypsy moth (Lymantria dispar), an invasive pest spreading across eastern North America. Also see the recent studies for competition model with the Allee efect [12–15]. Despite the history of the Allee efect in mathematical biology, the Allee efect is broadly discussed in [16–21]. Variation of the Allee efect as well as growing function exhibits ecological phenomena more precisely, especially when a plant or, animal, or organism is about to extinct. In 2016, Rocha et al. discussed the bifurcation and extinc- tion cases of species with a weak Allee efect, where the growth follows Richards’ growth law (details in [22]). Also see [23, 24] for fshery and genetic model with Allee efect and history within. Populations difer in the difusion strategies as they employ as well as in their environmental intensities [25–27]. Further, the competition between populations also has infuences on difusion [28–30]. In this paper, our main goal is to evoke the importance of Allee efect in reaction-dif- fusion equation and show how it efects populations in ecology. We study the impact of multiplicative Allee efect in classical difusion when the sparsity is either positive or negative. We show that negative sparsity implies weak Allee efect and the population survives in some domain and diverges otherwise. Also, positive sparsity gives strong Allee efect and the population extinct without any condition. Te stability conditions and the region of positive solution are presented. We present a number of numerical examples. Tis paper is organized as follows. In “Mathematical model” section, we present mathematical models and the fundamental assumption on per capita growth function, f(t, x, u). Allee efects without difusion are presented in “Allee efects without difusion” Keya et al. J Egypt Math Soc (2021) 29:4 Page 4 of 26 section, where in “Logistic model with Allee efects and harvesting” section we have dis- cussed the harvesting model without difusion.
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