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;
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• 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. Bifurcation point and steady state for the d 0 difusion coefcient, → with and without harvesting are presented in this portion. In “Allee efects with difusion” section we present main results of this paper. We pre- sent the existence of positive solution and the region where the solution exists for model (11). In “Examples and applications”section, we present some numerical examples where we present bifurcation points and regions of positive solution for particular functions. Finally, in “conclusion” section we present the conclusion and some open problems for further research.
Mathematical model Describing the Allee efect in a mathematical model is more challenging. We will use the form (1) for the logistic equation in cases where the coefcients are constant, but we want to consider the situations when the intrinsic growth rate r changes sign (concern- ing time, or in spatial models for space). du u ru 1 . dt = − K (1) Te assumptions that increase population density lead to decreases in the birth rate and increase the death rate may not always be valid. Allee (1931) observed that many animals engage in social behavior such as cooperative hunting or group defense that can cause their birth rate to increase or their death rate to decrease with population density [1]. Also, the rate of predation may decrease with prey density in some cases, as discussed by Ludwig in [31] (see recent studies on predation under the Allee efect in [42] and references therein). In the presence of such efects, which are typically known as Allee efects, the model (1) will take a more general pattern du uf (t, x, u), dt = (2)
where f(t, x, u) may be increasing for some values of u and decreasing for others. A sim- ple case of a model with an Allee efect is du ru(K u)(u M), dt = − − (3)
where, r > 0 and 0 < M < K . Te model (3) implies that the density of u will decrease if 0 < u < M or u > K but increase if M < u < K . Introducing difusion in this model is more challenging and efective in spatial heterogeneity. To demonstrate the model with difusion and the Allee efect, it is essential to intro- duce the following short description. Mathematically the growth rate per capita function u 0 f(t, x, u) will not have maximum value at = . If f(t, x, u) is negative when u is small, we defne such a growth pattern has a strong Allee efect or mandatory extinction. If f(t, x, u) is decreasing but still positive for the very low density of u, then this growth has a weak Allee efect or extinction-survival situation. Considering difusion in ecology model has open a new era. Keya et al. J Egypt Math Soc (2021) 29:4 Page 5 of 26
Let us now consider the general reaction-difusion mathematical model with homo- geneous Neumann boundary conditions ∂u d�u u(t, x)f (t, x, u), t > 0, x �, ∂t = + ∈ u n 0, t > 0, x ∂�, (4) u∇(0,·x) = u (x), x �. ∈ = 0 ∈ Rn ∂� C2 β Here d is the difusion rate. Te habitat is a bounded region in with ∈ + , β>0 and n is the outward normal vector. Te functions u(t, x) represent the population density of the species and it’s migration rate is positive, d > 0 . We assume that all the C1 β (�) β>0 x functions are in the class of + , for any ∈ . Te similar type model (4) was studied by Shi and Shivaji in [16] for weak Allee efects with global bifurcation analysis. Tey considered the following functions as a reaction term: u(t, x) A f (t, x, u) r(x) 1 , or f (t, x, u) m(x)u(t, x) b(x)u2(t, x), = − K(x) − 1 Bu(t, x) = − +
where the Allee efect is due to a type-II functional response. It is important to note that the function f(t, x, u) in (4) will be negative or positive such that the Allee efect will be seen depending on the choice of species mating dif- fculty or, low growth rate or high death rate as well as competition, etc. Tere are several kinds of growth function with the Allee Efect has been discovered since 1931. Tere are numerous example of f(t, x, u) with Allee efects, for examples: In 1988, Leah Edelstein-Keshet use a quadratic polynomial of the form [32]
f (t, x, u) r(x) b(u(t, x) a)2. = − − (5) Te negative quadratic Eq. (5) describes the per capita growth rate under the Allee efect. Here the function r(x) is the intrinsic growth rate of species and bounded. Also a, b > 0 , and depending on the choice of a there could be unconditional stability; extinc- tion-stability situation; and unavoidable extinction which we defned as no Allee efect (similar to logistic growth); weak Allee efect; and strong Allee efect. We will now defne an important function for further studies.
f R f (�) << � Defnition 1 A sparse function is a function : → such that | | , where is the magnitude of a set (i.e., the number of elements in the set). Te function f is | | 1 α also in the class of C + (�),0<α<1.
Biologically sparse means thinly scattered (species are set or planted here and there or, not being dense or close together), and any species of the population having this property is known as a sparse population. Tis idea leads to a sparsity function, which gives biological meaning to a mathematical expression. Te next two growth function with Allee efects are a variation of (5) and are obtained by modifying the Verhulst logistic model Keya et al. J Egypt Math Soc (2021) 29:4 Page 6 of 26
u(t, x) u(t, x) M(x) f (t, x, u) r(x) 1 − . = − K(x) K(x) (6)
u(t, x) u(t, x) f (t, x, u) r(x) 1 1 . = − K(x) M(x) − (7)
Te function (6) was broadly discussed in [33, 34]. Here M(x) is a sparse function, defned as M(x)
plausible relationships. On the other hand, the model is quite complicated, and conse- quently, much work has not been studied. Since each functions (5)–(10) are biologically meaningfull and generalized, see [41] for more details, here we mainly consider the function (7) with our difusion model (4). Using Eq. (7) is more appropriate to this paper since the main concept of this paper is to show both strong and weak Allee efect in a heterogeneous environment with and without difusion. Te sparsity function here gives the demographic Allee efect and made extinction of species within a short period in the presence of difu- sion. Te following sections will cover the analytic discussions and numerical approx- imations for further possibilities of this kind of Allee efect. By considering suitable sparse function and parameters the function of the other could give similar results and also broadly discussed and established in the references provided. Numerically we also consider the cases where functions (6) and (8) appears. Let us now consider the following mathematical model of (4) with f(t, x, u) as defned in (7). Ten we get the following regular (classical) difusion model with Allee efect: ∂u u(t, x) u(t, x) d�u(t, x) r(x)u(t, x) 1 1 , t > 0, x �, ∂t = + − K(x) M(x) − ∈ u n 0, � �� � t > 0, x ∂�, (11) ∇u(0,·x) = u (x), x �. ∈ = 0 ∈ For spatial positive functions M(x) and K(x), it is assumed that 0 < M(x)
u 0 f (t, , u) C1 β (�) β (0, 1) x (h1) For any ≥ , · ∈ + for ∈ , and for any ∈ , f (t, x, ) C1(R ) · ∈ + ; x u2(x) 0 f (t, x, u) 0 u(t, x)>u2(x) (h2) For any ∈ , there exists ≥ such that ≤ for , u2(x) K(x) x and ≤ for all ∈ ; and x u1(x) 0 f (t, x, ) (h3) For any ∈ , there exists ≥ such that · is increasing in 0, u (x) f (t, x, ) u (x), ) N > 0 [ 1 ] , · is decreasing in [ 1 ∞ , and there exists such that N f (t, x, u1) x ≥ for all ∈ .
From the assumptions, the function u2(x) indicates the crowding efects of the popu- lation at x, which may vary by location but it has a uniform upper bound, K(x). Te function u1(x) is where f(t, x, u) attains the maximum value. Here we still allow logis- u (x) 0 tic growth in cases where 1 = . Te constant N is the uniform upper bound of the growth rate per capita. Te function f(t, x, u) could be logistic or having Allee efect based on the following assumptions (see also Fig. 2):
f (t, x,0)>0 u (x) 0 f (t, x, ) 0, u (x) • Logistic type when , 1 = and · is decreasing in [ 2 ] (Fig. 2a). Keya et al. J Egypt Math Soc (2021) 29:4 Page 8 of 26
ab c 0.8 0.2 0.4 u 0.0 0.6 -0.2 0.3 0.4 -0.4 uf uf
uf 0.2 -0.6 -0.8 0.1 0.0 u -1.0 -0.2 -1.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 u 0.6 0.81.0
de1.2 f 0.8 u 0.0
0.9 0.4
-0.5 0.0 u f 0.6 f f -0.4 -1.0 0.3 -0.8
0.0 -1.2 -1.5 0.0 0.2 0.4 0.6 0.81.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u Fig. 2 The graphs on top row are growth rate uf(t, x, u), and the ones on lower row are growth rate per capita f(t, x, u), where a logistic for M 0 ; b weak Allee efect for M 1.1 cos(πx) ; c strong Allee efect → = + for M (1.1 cos(πx)) ; with K(x) 2.0 cos(πx) , d 1.0 , r(x) 1.2 over 0, 1 . In all fgure f(t, x, u) =− + = + = = =[ ] follows (7)
xs > 0 f (t, x,0) 0 x > xs • Weak Allee efect type when there exists such that ≥ for f (t, x,0) 0 x < xs s u (x)>0 and ≤ for in a non-empty open domain ⊂ . Also 1 , f (t, x, ) 0, u (x) f (t, x, ) u (x), u (x) · is increasing in [ 1 ] , and · is decreasing in [ 1 2 ] (Fig. 2b). f (t, x,0)<0 u (x)>0 f (t, x, u )>0 f (t, x, ) • Strong Allee efect type when , 1 , 1 , · is 0, u (x) f (t, x, ) u (x), u (x) increasing in [ 1 ] , · is decreasing in [ 1 2 ] (Fig. 2c).
Troughout the paper, we will consider f(t, x, u) as defned in (7), which satisfes (h1)– (h3). To estimate the long-time dynamics of the species whether they will persist or become extinct; steady state solution and eigenvalue-eigenfunction method are used to determine the dynamical behavior.
Allee efects without difusion Logistic model with Allee efects d 0 Consider the case when difusion is ignorable; that is → (for historical interest, see [42]), then the species birth (or death) is only dependent on growth rate (or death rate). If we allow Allee efects in this type of model, we get the following reduction model of d 0 (11) with = : du u(t) u(t) ru(t) 1 1 , t > 0, dt = − K M − (12) u(0) u , � �� � = 0 Keya et al. J Egypt Math Soc (2021) 29:4 Page 9 of 26
where M < K . Te parametric values, r, is defned as the intrinsic growth rate, K is the carrying capacity, and M is the Allee threshold. u(t) u(t) g(u) ru(t) 1 1 u Let = − K M − . To fnd the stationary solutions, ∗ of (12), du∗ 0 directly we set dt = such that
u∗ u∗ ∴ ru∗ 1 1 0 − K M − = u∗ 0, M, K . ⇒ = { } Also we get dg u u u u u u r 1 1 1 1 . du = − K M − + M − K − K M − (13) Since parameters r and K, except M, are strictly positive, and M is the Allee threshold (positive or negative threshold) and M < K , then depending on the choice of M, we have the following two cases: Case 1 When 0 < M < K (weak Allee efect): u 0, M, K Using ∗ = { } in (13), we get, dg (0) r < 0 du =− ; dg M (M) r 1 > 0 and du = − K ; dg K (K) r 1 < 0. du =− M − u 0, K u M Terefore, ∗ = { } are stable solution of (12) and ∗ = is the unstable state of (12) when the species are facing weak Allee efects. Case 2 When M < 0 < K (strong Allee efect): u 0, M, K M M Putting ∗ = { } in (13), where = −| | , we get that the equilibrium 0 M M does not change its stability. But when, = −| |, dg dg M (M) ( M ) r 1 | | > 0 and du =du −| | = + K ; dg K (K) r 1 > 0. du = + M | | u 0 u M, K Terefore, ∗ = is the only stable solution of (12) and ∗ = are unstable state of (12), when the species are facing strong Allee efects. For a single species population in ecology, the biological signifcance of these sta- ble-unstable strategies is that either the species repelled or converge to the carry- u 0 ing capacity. Te present study employs that the stable situation is either = or u(t) K = , i.e., the population will die out or survive, respectively, and the maximum population density is K. A bifurcation point or threshold level can be found in an u(t) M unstable solution, = . For the sake of diversity and to cover the wide range of studies, in the following sec- tion, we introduce the harvesting term in absence of difusion. Keya et al. J Egypt Math Soc (2021) 29:4 Page 10 of 26
Logistic model with Allee efects and harvesting If harvesting is involved then the model can be formulated in the following way from (12): du u(t) u(t) ru(t) 1 1 Hu(t), t > 0, dt = − K M − − (14) u(0) u � �� � . = 0 H 0 H < r Here the harvesting function H is constant, ≥ and , and in case of Allee efects, M < K . Let us defne the following function u(t) u(t) g∗(u) u(t) r 1 1 H , = − K M − −
and dg u u u u u u ∗ r 1 1 1 1 H. du = − K M − + M − K − K M − − (15) du 0 Now letting dt = , we get u u u r 1 1 H 0 − K M − − = ru2 r(K M )u KM (r H) 0 and u 0 ⇒ − + + + = ; = r(K M) r2(K M)2 4rKMH u + ± − − and u 0. ⇒ = 2r ; =
Terefore, the three equilibrium states of (14) are u 0, 1 = 1 u r(K M) r2(K M)2 4rKMH , and 2 = 2r + + − − 1 2 2 u3 r(K M) r (K M) 4rKMH . = 2r + − − − Te equilibrium states u2 and u3 exist as long as
r2(K M)2 4rKMH > 0. − − D r2(K M)2 4rKMH u u Let us defne := − − which modify 2 and 3 in the following mathematical structures 1 D u (K M) , 2 =2 + + r 1 D u3 (K M) . =2 + − r
Now from Eq. (15), we get
dg∗ (u 0) (r H)<0, since r > 0, H 0. du 1 = =− + ≥ Keya et al. J Egypt Math Soc (2021) 29:4 Page 11 of 26
u 0 Tus we conclude that 1 = is a stable solution either there is a strong Allee efect or weak Allee efect. Te rest two equilibria states are u2 and u3 , and in terms of H and M, the following two situations arise: When 0 < M < K , i.e., weak Allee efect, then we get the following three cases in terms of H:
r(K M)2 r(K M)2 r(K M)2 H < − , H − , and H > − . 4KM = 4KM 4KM r(K M)2 H > − D2 < 0 u , u C R Case I When 4KM , that is if then 2 3 ∈ \ hence their stability is irrelevant and we have no conclusion. r(K M)2 H − D 0 Case II If = 4KM then = and the system (14) has one identical solu- u u K M tion, 2 = 3 = +2 and from (15) it is seen that
dg∗ K M + 0. du 2 =
Terefore, in this situation, the system has a unique positive equilibrium, which is semi-unstable. r(K M)2 H < − D > 0 Case III If 4KM then and the problem (14) has two distinct solu- tions u2 and u3 , where 1 D 1 D u (K M) and u3 (K M) . 2 = 2 + + r = 2 + − r
Since D > 0 , u2 is always positive. We recall (15) and get the simplifed form as follows: dg 3 2(K M) ∗ r u2 + u 1 H, du = −KM + KM − −
such that dg 3 2(K M) ∗ (u ) r u2 + u 1 H. du 2 = −KM 2 + KM 2 − − u 1 (K M) D Using 2 = 2 + + r , we get dg (K M)2 K M 3 ∗ (u ) r + 1 H + D D2 . du 2 = 4KM − − − 2KM + 4rKM (16) dg ( )2 ( )2 ∗ K M r K M 1 H 0 H r K M It is noted that du +2 = 4+KM − − = when = 4KM− . Hence there exits at least one set of positive parameters, r, K, M, and H with M < K ( )2 ( )2 H r K M ǫ, ǫ>0 H < r K M ǫ such that we assume = 4KM− − for 4KM− , where is small D 2√rǫKM > 0 enough. Using the value of H, it is found that = . Substituting H and D in (16) yields Keya et al. J Egypt Math Soc (2021) 29:4 Page 12 of 26
dg∗ K M 3 (u2) r ǫ + 2√rǫKM 4rǫKM du =− + − 2KM + 4rKM K M r ǫ + √rǫKM 3ǫ =− + − KM − (K M)√rǫKM r 2ǫ + < 0. =− − − KM
In case of equilibrium point u3 , it is seen that u3 is positive only for r(K M)>D . Ten dg ( )2 + ∗ K M 0 H < r K M using the fact that, du +2 = and 4KM− and from the above con- dg ∗ structive manner, we found that du (u3)>0. All the results from case III are summarized as follows:
dg∗ K M D > 0 u2 > 0 + 0 1. For , it is obvious that . Using the fact that du 2 = and ( )2 dg∗ H < r K M (u )<0. 4KM− , we have du 2 dg∗ r(K M)>D u > 0 (u )>0. r(K M) Te stable and convergent solutions are presented in Fig. 3 both for weak and strong u(t) u Allee efects. For large time, t, the solution → 2 when M is positive (Fig. 3a). u 0 Similarly, in case of strong Allee efect, the solution u(t) is converging to 1 = (Fig. 3b). u 0, u In conclusion, for weak Allee efect, we have two stable equilibria 1 = { 2} , and one unstable equilibrium state u3 . Hence the solution of the model (14) is decreasing for 0 < u(t) a 0.9 b 0.5 0.4 0.8 0.3 0.7 Density Density 0.2 0.6 0.1 0.5 0 0 5 10 15 20 0 5 10 15 20 time t time t Fig. 3 Numerical solutions of (14) for r 0.8147, K 1, u0 0.5 with a 0 < M 0.30 < K, H 0.2661 , = = = = = and a M 0.9058 < 0 < K, H 0.1270 =− = Keya et al. J Egypt Math Soc (2021) 29:4 Page 13 of 26 u 0, 1 = 1 D∗ u (K M ) , and 2 =2 −| | + r 1 D∗ u (K M ) . 3 =2 −| | − r D r2(K M )2 4rK M H Where ∗ := +| | + | | . Here we also get three cases in terms of H: r(K M )2 r(K M )2 r(K M )2 H < − +| | , H − +| | , and H > − +| | . 4K M = 4K M 4K M | | | | | | r(K M )2 D H < − +| | Since ∗ is not valid for this 4K M , so there is no equilibrium point, that is C R | | u2, u3 . ∈ \ r(K M )2 +| | r, K On contrary, the value − 4K M is strictly negative since and |M| are strictly | | r(K M )2 r(K M )2 H − +| | H > − +| | positive. Hence H is negative for either, = 4K M or, 4K M , which | | | | is a contradiction of our earliest assumption that harvesting can not be negative. Consider- ing this cases, we can conclude that there is no condition of D∗ found, such that a positive equilibrium exists when the harvest is positive. Terefore, if we allow harvesting in a strong Allee efect, the resulting behavior is diverg- u 0 ing, and does not exist a stable point except possibly, 1 = . Let us now introduce the difusion term in our considered model for further study. Allee efects with difusion In the dynamical behavior of (11) without an Allee efect, the results are well established. When the species has logistic growth, there is a critical value d1 > 0 such that 0 < d < d1 , i.e. for slow difusion, there is a unique positive steady state u∗(x) , which is the asymp- u 0 totic limit for any non-negative initial distribution except 0 ≡ , thus the persistence of the population is achieved. If d > d1 , the difusion is fast and the only non-negative steady u 0 state solution is ∗ = , thus the extinction is inevitable; the readers can check the literature ( [16], Section 1) for more details about the inevitable extinction. If u(x) is any stationary solution of (11), consider the following eigenvalue problem to analyze the linear stability of the system, �φ σ r(x)φ f ufu 0, x �, φ 0, x ∂�, + [ + ]= ∈ ∇ = ∈ where fu is the derivative of f with respect to u, and u∗(x) u∗(x) f (t, x, u) r(x) 1 1 . = − K(x) M(x) − Te function φ(x) is the eigenfunction and σ is the corresponding eigenvalue. Here the d 1/σ (f , �) σ (f , �) constant 1 = 1 , and 1 is the principle eigenvalue of the linearized sys- tem of u∗(x) u∗(x) �φ σ r(x)φ 1 1 0, x �, φ 0, x ∂�. + − K(x) M(x) − = ∈ ∇ = ∈ (17) Keya et al. J Egypt Math Soc (2021) 29:4 Page 14 of 26 Bifurcation analysis σ d 1 u (x) Let = − and ∗ be the stationary solution of (11) then u∗(x) u∗(x) �u (x) σ r(x)u (x) 1 1 0, x �, ∗ + ∗ − K(x) M(x) − = ∈ � �� � (18) u∗ n 0, x ∂�, u∇(0, x·) =u (x), x ∈ �. = 0 ∈ When a stable solution of (18) is viewing as an equilibrium solution to (11) can be deter- mined by the eigenvalue problem: u∗(x) u∗(x) �φ σ r(x)φ 1 1 0 + − K(x) M(x) − = u∗(x) u∗(x) φ 2 dx σ r(x)φ2 1 1 dx 0 ⇒− |∇ | + − K(x) M(x) − = � � φ 2 dx |∇ | σ � . ⇒ = ( ) ( ) r(x)φ2 1 u∗ x u∗ x 1 dx − K(x) M(x) − � Hence the principle eigenvalue is defned by 2 u∗ dx � |∇ | σ1 sup . = 1 u∗(x) u∗(x) u H0 (�) r(x)u 2(x) 1 1 dx ∈ ∗ − K(x) M(x) − � Tus the bifurcation point for positive solutions is defned by 1 2 2 d1 sup f (t, x, u∗)u∗ (x) dx u∗ dx 1 . (19) = σ1(f , �) = u H1(�) : |∇ | = ∈ 0 � � f (t, x, u ) σ σ (f , �) Here ∗ follows assumptions (h1)–(h3) and 1 ≡ 1 is a bifurcation point, where nontrivial solution of (18) bifurcate from trivial solution. Te classical type of bifurcation diagram at σ1 is shown in Fig. 4. Te more details about this type of bifurca- tion can be found in ([16], Section 2). Existence of a positive stationary solution Consider the steady state of (11), if u∗(x) being the positive solution then we get u∗(x) u∗(x) 0 d�u∗(x) r(x)u∗(x) 1 1 , x �, = + − K(x) M(x) − ∈ which implies 1 u∗ u∗ �u ru∗ 1 1 , x �, − ∗ = d − K M − ∈ � �� � (20) u∗ n 0, x ∂�, ∇u (0,·x) = u (x), x ∈ �. ∗ = 0 ∈ Keya et al. J Egypt Math Soc (2021) 29:4 Page 15 of 26 Fig. 4 Classical bifurcation diagram from left to right showing for logistic; strong Allee efect and weak Allee efect, respectively. Here σ d 1 , where d is the difusion constant = − Now the following situations occur for a steady-state solution. 2 (K1 M1) Teorem 1 Let σ1 r1 − . Ten (20) has no positive solution. ≥ 4K1M1 Proof σ1 Let be the principle eigenvalue of the operator − with Neumann boundary conditions and φ>0 in be a corresponding eigenvector. Assume that u be a positive solution of (20) then by the Green’s identity, we have Keya et al. J Egypt Math Soc (2021) 29:4 Page 16 of 26 (u�φ φ�u) dx 0 − = � 1 u(t, x) u(t, x) r(x)u(t, x) 1 1 φ σ1φu dx 0 ⇒ d − K(x) M(x) − − = (21) � 1 u(t, x) u(t, x) φu(t, x) r(x) 1 1 σ dx 0. ⇒ d − K(x) M(x) − − 1 = � u u f (u) r1 1 1 Since f(t, x, u) follows assumption (h1)–(h3), let ˜ = − K M − , where 1 1 r1, K1 are maximum value of the functions r(x), K(x) and M1 is the minimum value of M(x), respectively in the domain . Ten ∂f (u) K M 2 ˜ r 1 + 1 u . ∂u = 1 K M − K M 1 1 1 1 ∂f (u) ˜ 0 u K1 M1 For critical value we know ∂u ≡ , which gives = +2 . ∂2f (u) 2r ˜ 1 < ( ) Since 2 0 , the function f u has a maximum value when ∂u =−K1M1 ˜ K M u 1 + 1 = 2 . Tus f0 sup f (u) ˜ := � ˜ K M K M r 1 1 + 1 1 + 1 1 = 1 − 2K 2M − 1 1 K M K1 M1 r 1 − 1 − = 1 2 · 2 2 (K1 M1) r1 − , = 4K1M1 u(t, x)f (t, x, u) u(t, x)f (t, x, u) u(t, x) 0 and ≤ ˜ for ≥ . Tus Eq. (21) implies 1 0 (u�φ φ�u) dx φu f (t, x, u) σ1 dx = − ≤ d ˜ − � � 1 (22) ∴ φu f (t, x, u) σ dx 0. d ˜ − 1 ≥ � 2 (K1 M1) But if σ1 r1 − then ≥ 4K1M1 1 1 φu f (t, x, u) σ dx < φu f σ dx 0, d ˜ − 1 d ˜0 − 1 ≤ � � which is a contradiction2 of (22). Hence (20) has no positive solution for (K1 M1) σ1 r1 − . ≥ 4K1M1 Keya et al. J Egypt Math Soc (2021) 29:4 Page 17 of 26 M0, r1 K0 µ (0, 1 Teorem 2 Tere exists positive constants , ˜ and ∈ ] such that for K0 > K0 , (20) has a positive solution and the positive solution occurs in the region K ˜K 0 µ2, 0 2 2 . Proof At frst we will prove an existence result for (20). Let σ be the principal eigenvalue of the φ>0 operator − with Neumann boundary conditions and is the corresponding eigen- φ 1 δ>0, µ (0, 1 m > 0 function such that || ||∞ = . Hence there exists ∈ ] and such that 2 2 φ σφ m for x �δ, |∇ | − ≥ ∈ (23) φ µ for x ∂�δ, ∇· ≥ ∈ (24) �δ x � d(x, ∂�)<δ where := ∈ | . From (20), we have the reaction function as defned in a simplest form such that u u g(u) r u 1 1 , = 1 − K M − 0 0 K K M M K1, M1 and r1 where 0 ≤ 1 , 0 ≤ 1 and are the maximum value of K(x), M(x) and r(x) , respectively as defned in Teorem 1. It is easily observed that the zeros of g are 0, M0 and K0 where K0 > M0 and hence r1 g(u) u(u K0)(u M0). := −K0M0 − − ∂g u Let ∗ be the frst positive zero of ∂u in which 1 2 2 K0 M0 u∗ (K0 M0) K M K M < + . = 3 + − 0 + 0 − 0 0 3 K M 0, 0 + 0 But g is convex on 3 and hence r1 2 2 inf g(u)< (K0 M0) K0 M0 K0M0 r1u∗. := − u 0,K0 3 + − + − = ∈[ ] If we recall the second set of Fig. 2, which describes the behavior of u and u∗ (Fig. 5). We frst claim that 1 r1 2 2 < (K0 M0) K0 M0 K0M0 K0 K0 3 + − + − r M 1 0 . = (K M ) K 2 M2 K M 0 + 0 + 0 + 0 − 0 0 (1) Tus 0 as K0 tends to infnity. Hence there exists K0 such that for every K0 → (1) K0 > K0 , we have Keya et al. J Egypt Math Soc (2021) 29:4 Page 18 of 26 0.0 0.2 0.4 0.6 0.8 1.0 0.8 u 0.6 0.4 uf 0.2 0.0 u* M0 K0 -0.2 -0.4 Fig. 5 Growth rate function uf(t, x, u) with weak Allee efect for M 1.1 cos(πx) , K(x) 2.0 cos(πx) , = + = + d 1.0 and r(x) 1.2 over the domain 0, 1 = = =[ ] m > . (25) K0 K0 (2) Next, we also claim that as K0 . Hence there exists K0 such that for M0 →∞ →∞ (2) every K0 > K0 , we have K K 0 µ2, 0 (M , K ), 2 2 ⊂ 0 0 (26) Kµ inf g(u)>0 and K K . Finally := u 0 µ2, 0 ∈ 2 2 Kµ 1 K K min g 0 µ2 , g 0 K = K 2 2 0 0 2 2 1 r1µ 2 K0µ r1 K0 min K0(µ 2) M0 , K0 M0 = K − 4M − 2 − 4M 2 − 0 0 0 r µ2 K µ2 r K min 1 (µ2 2) 0 M , 1 0 M , = − 4M − 2 − 0 4M 2 − 0 0 0 (3) (2) which tends to infnity as K0 . Tus there exists K0 > K0 such that for every (3) →∞ K0 > K0 , we have Kµ σ< . K0 (27) K max K (1), K (3) M > 0, r > 0 K > K Defne ˜0 := 0 0 and for some 0 1 and 0 ˜0 , (20) has a positive solution. At this stage, we will use the upper and lower solution method to show K K 0 µ2, 0 µ (0, 1 the existence of a positive integral which lies in the interval 2 2 ; ∈ ]. K0 2 ψ φ x �δ First, we prove that = 2 is a sub-solution of (20). For ∈ , Keya et al. J Egypt Math Soc (2021) 29:4 Page 19 of 26 �ψ K (σ φ2 φ 2) − = 0 − |∇ | K m using (23) ≤− 0 ; using (25) ≤− ; inf g(u) ≤ u 0,K ∈[ 0] u u r u 1 1 . ≤ 1 − K M − 0 0 Also for the boundary: �ψ K (σ φ2 φ 2) − = 0 − |∇ | K σ ≤ 0 Kµ inf g(u) using (27) K K ≤ = u 0 µ2, 0 ; ∈ 2 2 u u r u 1 1 . ≤ 1 − K M − 0 0 K ψ 0 φ2 φ K Hence = 2 is a sub solution of (20). We also note that = 0 is a super solution for (20) since K(x) is the carrying capacity of the environment, therefore the solution would not exceed it. However, if the solution exceeds the carrying capacity for some- times, the solution will not be stable, and within a short period, it will again converge to K K 0 µ2, 0 K(x). Tus there exists a positive solution of (20) in 2 2 . Examples and applications To learn the importance of application of Allee efects in biological invasions, we can consider two situations: In one case, consider a species, which is in red list, that is the population density is very low, so there is not enough mating and the environment is not favorable, so ultimately in a short time, the species will go extinct. But if we can improve the sparse function by giving their favorable sparse (in this case, the sparse function will be very low), then the species could survive. In other cases, if a species is invaded by another species (e.g., bacteria or virus), then the sparse function for invader species will play an important role to immune the invaded species. In this situation, we have to higher the parametric value of sparse func- tion so the invader species could not fnd favorable space to spread in invaded species. Consequently, the invader species will go extinct and the invaded body will be immune. Considering all types of similar cases and form the numerical assumptions, the follow- ing numerical behaviors are simulated. Example 1 K 2.5 u 0.5 Consider the carrying capacity = with an initial density of any species 0 = and r 1.0 intrinsic growth rate of the species, = . Ten the numerical behavior of solutions of (12) shows that if the species has a strong Allee efect and the sparse function M is nega- tive, then the solution will always be repelled to zero, which is following in Fig. 6a. Keya et al. J Egypt Math Soc (2021) 29:4 Page 20 of 26 ab 0.5 2.5 M=−1.5 M=0.1 M=−1.0 M=0.2 0.4 M=−0.5 2 M=0.5 M=−0.2 M=1.0 0.3 M=−0.1 1.5 M=1.5 Density 0.2 Density 1 0.1 0.5 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 time t time t Fig. 6 Solutions of (12) with respect to time t, for K 2.5 , u0 0.5 and r 1.0 where in (a) = = = M 1.5, 1.0, 0.5, 0.2, 0.1 and in (b) M 1.5, 1.0, 0.5, 0.2, 0.1 =− − − − − = a b 2.0 M 0.6 1.5 0.9 u 1.2 1.8 ity of 1.0 ns De 0.5 K=2.0 0.0 0 5 10 15 time t Fig. 7 Solutions of (12) with respect to time t, where r 1.0 and in (a) K 2.5 , M 1.15 and = = = u0 0.5, 0.9, 1.15, 1.5 and in (b) K 2.0 , u0 0.9 and M 0.6, 0.9, 1.2, 1.8 = = = = If the species are following a weak Allee efect and the sparse function M is positive but M < K M (0, u ) , then we see that for any ∈ 0 , the solution of (12) converges to its maxi- mum yields, towards carrying capacity. But if u0 < M < K , then the solution is repelling M u and converging to zero. For = 0 , there exists a constant solution. See Fig. 6b which follows our assumption (h1)–(h3) and the stable conditions of (12). Example 2 When difusion is ignorable, that is in the model (12) we have two stable solutions, one is repelling and the other one is asymptotically stable. In numerical behavior, we see simi- lar results where either the species is extinct or survives. In numerical behavior at frst K 2.5 r 1.0 we vary initial density, and see that the solution, u(t) of (12) for = , = and M 1.15 = either tends to zero or, converge to K. So the solutions of (12) are locally stable. u M Also there is a constant solution if 0 ≡ as seen in Fig. 7a. In Fig. 7b, we exhibit that if sparse function M is having lower signifcance and the ini- tial population is in a better position then the species will exist and will converge to K. Keya et al. J Egypt Math Soc (2021) 29:4 Page 21 of 26 Contrariwise if the sparse function is more signifcant in the domain and initial densities are very low, then the species will die out in a short time. In Fig. 7b we vary sparse func- M 0.6, 0.9, 1.2, 1.8 u0 0.9 K 2 tion as = { } with = , = and growth rate is same as above. Here we also see one stable solution and one repelled solution. Example 3 Consider the Allee efect model with harvesting as in (14). Here we see that for a small harvesting rate and sparse function, positive and lower than initial density gives stability at its utmost point. And on the contrary situation, the solutions converge to zero. Again we consider various harvesting rates; lower or higher than initial density for strong Allee efects, that is the sparse function is being negative. In this case, zero is the only converging solution. Te following Fig. 8 is showing the signifcance of harvesting K 4.0 u 1.0 r 1.0 M 0.6 functions while we choose = , 0 = , = and for = (Fig. 8a) and for M 0.6 =− (Fig. 8b). Tis is also a case of local stability and both fgure shows the condi- tions of Allee efects as described for model (14). Example 4 K(x) 2.5 cos(πx) Assume now that difusion is not ignorable, consider in (11), = + and M(x) 1.2 cos(πx) d 0.1 r 1.0 = + with migration rate = , growth rate = and initial den- u 0.5 � (0, 1) sity 0 = . Ten the model (11) has positive solution in the domain = , which is shown in Fig. 9. a 4 b 1 H=0.1 H=0.1 H=0.3 0.8 3 H=0.3 H=0.7 H=0.7 H=1.0 H=1.0 0.6 2 Density Density 0.4 1 0.2 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 time t time t Fig. 8 Solutions of (14) with respect to time t for K 4.0 , u0 1.0 and r 1.0 with various harvesting rate as = = = H 0.1, 0.3, 0.7, and 1.0 ; where a M 0.6 (weak Allee efect), and b M 0.6 (strong Allee efect) = = =− Keya et al. J Egypt Math Soc (2021) 29:4 Page 22 of 26 a 1.6 b 1.1 u(50,x) 1.4 1.0 ) 0.9 1.2 0.8 u(t,x) 1.0 ns ity u(50,x 0.7 De 0.8 Ave. Density u(t,x) 0.6 0.6 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 Domain time Fig. 9 a Solutions of (11) with respect to space at t 50 and b average solutions of (11) with respect to time = t for K(x) 2.5 cos(πx) , M(x) 1.2 cos(πx) and r(x) 1.0 on � (0, 1) , with u0 0.5 , d 0.1 = + = + = = = = a d= b 1.8 d= 2.1 0.0 0.0 0.001 0.001 1.8 0.100 1.5 0.100 0.153 0.153 0.154 0.154 1.5 1.2 1.2 0.9 ns ity De 0.9 Ave. Density 0.6 0.6 0.3 0.3 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0 9 18 27 36 45 Domain time Fig. 10 Dynamical behavior of solutions of (11) for various difusion coefcients at t 50 in (a) and average = solutions of (11) in (b) for K(x) 2.5 cos(πx) , M(x) 1.2 cos(πx) and r(x) 1.0 on � (0, 1) , with = + = + = = initial density u0 0.5 = Example 5 While considering difusion is not zero and not close to zero if we choose some particu- d d d lar function then we see that there exists one ∗ for which, when ≥ ∗ there exist no positive solution, positive solutions occur only on 0 < d < d∗ . Particularly in (11) let K(x) 2.5 cos(πx) M(x) 1.2 cos(πx) r(x) 1.0 u 0.5 = + , = + , = and 0 = . Ten numeri- d 0.154 cally we get a bifurcation point ∗ = . Any value greater than this d∗ is always tending to zero, and the species extinct after a short time. Te following Fig. 10 is showing the dynamical behavior of d∗ , that is when difusion or migration is very slow, the species will persist. Tis numerical behavior is analytically proven in Teorem 1. Keya et al. J Egypt Math Soc (2021) 29:4 Page 23 of 26 Example 6 K(x) 2.5 cos(πx) For model (11), let the distribution functions are = + , M(x) 1.2 cos(πx) r(x) 1.0 � (0, 1) = + , growth rate is = on the domain = and initially u 0.5 crowding density is 0 = . 0, K0 K0 µ2, K0 µ (0, 1 Ten the positive solutions of (11) lies on the region [ 2 ]⊂ 2 2 ; ∈ ] , which is analytically asserted in Teorem 2. In numerical assertion, from Fig. 11 when d 0.05 0.4, 0.9 0, K0 d 0.1 = (Fig. 11a), region of positive solution is [ ]⊂[ 2 ] ; when = 0.4, 1.2 0, K0 d 0.154 (Fig. 11b), region of positive solution is [ ]⊂[ 2 ] and when = (Fig. 11c), 0, 0.4 0, K0 d 0.154 d region of positive solution is [ ]⊂[ 2 ] . Here = = ∗ is a bifurcation point as described in Fig. 10. Example 7 Considering difusion model (4) with growth rate function with Allee efect as in (6), then the behavior of solution diverges for certain d > d∗ . If we increase the initial value then d∗ changes for the same sparse function and carrying capacity. For an initial ab 1.3 1.3 u(t,x) K(x)/2 1.0 1.0 u(t,x) of 0.8 0.8 u(t,x) ave. ave. of u(t,x) K(x)/2 0.5 0.5 0 5 10 15 20 0 5 10 15 20 25 time t time t c 1.2 1.0 u(t,x) x) K(x)/2 0.8 u(t, 0.6 of 0.4 ave. 0.2 0.0 0 5 10 15 20 time t Fig. 11 Region of positive solutions of (11) at t 50 for difusion coefcients a d 0.05 , b d 0.1 , and = = = c d 0.153 , where K(x) 2.5 cos(πx) , M(x) 1.2 cos(πx) and r(x) 1.0 on � (0, 1) , with initial = = + = + = = density u0 0.5 = Keya et al. J Egypt Math Soc (2021) 29:4 Page 24 of 26 d 0 < d d value ∗ is not unique and each cases solution exists for ≤ ∗ . In (4) with (6), let K(x) 2.6 cos(πx) M(x) 1.1 cos(πx) r(x) 1.0 = + , = + , = . Ten numerically we get a d 0.021 u 0.5 u 0.9 bifurcation point ∗ = for 0 = , see Fig. 12a. For 0 = , we get bifurcation d 0.038 point ∗ = as in Fig. 12b. Te numerical illustrations also indicate that there is an impact of initial density when the Allee efect occurs. Te simulating result ensures the theoretical outcome as shown in Teorem 1. Conclusion In this paper, we studied a generalized reaction-difusion model for single species hav- ing an Allee efect, where the species has carrying capacity, K, growth rate function, r and sparse function, M,0< M < K . For this generalized model, we discussed various Allee efects, and regarding these Allee efects, we discussed diferent types of growth function and their behavior. While considering a particular growth with the Allee efect (either strong or, weak, or both), we frst considered difusion is ignorable and found the d 0 bifurcation point and stable steady state when → . We also fnd a stable state involv- d 0 d ing harvesting. When ≡ and not close to zero, we have found a bifurcation point ∗ for which the steady states changes to unsteady. For some particular function, we see d d d that there exists one ∗ for which when ≥ ∗ there exists no positive solution, positive 0 < d d d solutions occur only on ≤ ∗ . For any greater value of ∗ , the solution tends to zero and the species is in extinction at a short time. Te Fig. 10 is showing the dynami- cal behavior of d∗ as shown analytically in Teorem 1. Te existence of a solution (might have multiple solutions) is proved and the region of the positive solution is also found. Some numerical examples were presented to justify the analytic study in a non-empty open domain. Finally, let us outline some open problems and topics for research and discussion 1. Consider (11) with the Dirichlet and the Robin boundary conditions. 2. For variable periodic parameters, does a positive periodic solution depend on the dif- fusion coefcient? ab 2.0 d= 0.0 0.01 1.5 0.021 0.022 u0 = 0.5 ity ns 1.0 De 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Domain Fig. 12 Dynamical behavior of solutions of (11) for various difusion coefcients at t 200 in (a) solutions = of (4) for u0 0.5 (b) solutions of (4) for u0 0.9 with K(x) 2.5 cos(πx) , M(x) 1.2 cos(πx) and = = = + = + r(x) 1.0 on � (0, 1) = = Keya et al. J Egypt Math Soc (2021) 29:4 Page 25 of 26 3. Originally some of the models described in terms of ordinary diferential equa- tions incorporate some kind of delay, for example, the Nicholson’s blowfies equa- tion. Extend the results of the present paper to delay models, explore the infuence of delays and compare results with classical and directed difusion strategies. 4. It is known that, for example, for the logistic growth with a periodic carrying capac- ity, the average of the positive periodic solution over the time cycle is less than the average resource function. Is this true for the periodic model with the directed difu- sion? 5. Consider the modifed model of (11) as ∂u u(t, x) u(t, x) u(t, x) d� r(x)u(t, x) 1 1 , t > 0, x �, ∂t = P(x) + − K(x) M(x) − ∈ (u/P) n� 0, t >� 0, � �� � x ∂�, u∇(0, x) · u=(x), x ∈ �. = 0 ∈ (6.1) where P(x) is the resource distribution function of species u(t, x), and other func- tions have the same interpretation as noted for (11). Acknowledgements The author gratefully acknowledges the anonymous referees for their insightful and constructive comments and sug- gestions. The authors are also grateful to Professor Jim Michael Cushing for his constructive suggestions on the previous versions of the manuscript for fruitful discussions during ICMA VII conference meeting, 2019 at Arizona State University. The author M. Kamrujjaman research was partially supported by the Ministry of Science and Technology (MOST), Grant BS193 for year 2019-2020, Bangladesh. Authors’ contributions KNK and MK derived the mathematical model, designed the study and the frst draft of the manuscript. KNK and MK car- ried out the theoretical analysis and numerical simulations. MK and MSI provided the literature review and fnal drafting. This work was carried out in collaboration among all authors. All authors read and approved the fnal manuscript. Funding Not applicable. Availability of data and materials Not applicable. Competing interests The authors declare that there is no confict of interest regarding the publication of this paper. 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