Dynamics of a predator-prey system with prey subject to Allee effects and disease

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1 Dynamics of a predator-prey system with prey subject to Allee effects and disease 3 Yun Kang, Sourav Kumar Sasmal,, Amiya anjan Bhowmick, 3, Joydev Chattopadhyay, Abstract n this article, we propose a general predator-prey system with prey subject to Allee effects and disease with the following unique features: i) Allee effects built in the reproduction process of prey where infected prey -class) has no contribution; ii) Consuming infected prey would contribute less or negatively to the growth rate of predator -class) in comparison to the consumption of susceptible prey S-class). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model S-Allee Model) as well as its corresponding model in the absence of Allee effects S-no-Allee Model); we obtain the complete dynamics of both models: a) S-Allee Model may have only one attractor extinction of all species), two attractors bi-stability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors tri-stability); b) S-no-Allee Model may have either one attractor only S-class survives or the persistence of S and -class or the persistence of S and -class) or two attractors bi-stability with the persistence of S and -class or the persistence of S and -class). One of the most interesting findings is that neither models can support the coexistence of all three S,, -class. This is caused by the assumption ii), whose biological implications are that and -class are at exploitative competition for S-class whereas -class cannot be superior and -class cannot gain significantly from its consumption of -class. n addition, the comparison study between the dynamics of S-Allee Model and S-no-Allee Model lead to the following conclusions: ) n the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; ) n the presence of Allee effects, disease may be able to save prey from the predation-driven extinction and leads to the coexistence of S and -class while predator can not save the disease-driven extinction. All these findings may have potential applications in conservation biology. Key words: Allee Effect, Functional esponses, Disease/redation-Driven Extinction, Bi-stability, Tri-stability, Eco-epidemiological System ntroduction Allee effects, referred to a biological phenomenon characterized by a positive correlation between the population of a species size or density and its per capita growth rate at its low population sizes/densities [, 56, 49], have great impacts in species establishment, persistence, invasion [3, 76, 4, 7, 6, 9, 49, 44] and evolutionary traits []. Empirical evidence of Allee effect has been reported in many natural populations including plants [5, 9], insects [5], marine invertebrates [67], birds and mammals []. Various mechanisms at low population sizes/densities, such as the need of a minimal group size necessary to successfully raise offspring, produce seeds, forage, and/or sustain predator attacks, have been proposed as potential sources of Allee effects [4, 53, 6, 9, 65, 66, 6]. ecently, many researchers have studied the impact of Allee effects on population interactions [e.g., see [6, 8, 43, 5, 75, 45, 46, 44] as well as the interplay of Allee effects and disease on species s establishment and persistence [38, 79, 37, 7, 47, 48]. All these research suggest the profound effects of Allee effects in population dynamics, especially when it couples with disease. Science and Mathematics Faculty, Arizona State University, Mesa, AZ 85, USA. yun.kang@asu.edu Agricultural and Ecological esearch Unit, ndian Statistical nstitute, 3, B. T. oad, Kolkata, 78, NDA. reprint submitted to Elsevier sourav.sasmal@gmail.com 3 Agricultural and Ecological esearch Unit, ndian Statistical nstitute, 3, B. T. oad, Kolkata, 78, NDA. amiya r@isical.ac.in; amiyaiitb@gmail.com 4 Agricultural and Ecological esearch Unit, ndian Statistical nstitute, 3, B. T. oad, Kolkata, 78, NDA. joydev@isical.ac.in

2 Eco-epidemiology is comparatively a new branch in mathematical biology which simultaneously considers the ecological and epidemiological processes [5]. Hadeler and Freedman [33] first introduced a eco-epidemiological model regarding predator-prey interactions with both prey and predator subject to disease. Since the work of Hadeler and Freedman 989), the research on eco-epidemiology as well as its biological importance has gained great attention [6, 7, 8, 3, 7, 3, 5, 78, 4, 73, 35, 39, 5, 69, 68]. Many species suffer from Allee effects, disease and predation. For instance, the combined impact of disease and Allee effect has been observed in the African wild dog Lycaon pictus [, ] and the island fox Urocyon littoralis [7, 4]. Both the African wild dog and island fox should have their enemies in the wild. Thus, understanding the combined impact of Allee effects and disease on population dynamics of predator-prey interactions can help us have better insights on species abundance as well as the outbreak of disease. Therefore, we can make better policies to regulate the population and disease. Thus, for the first time, we propose a general predator-prey model with Allee effects and disease in prey to investigate how the interplay of Allee effects and disease in prey affect the population dynamics of both prey and predator. More specifically, we would like to explore the following ecological questions:. How do Allee effects affect the population dynamics of both prey and predator?. Which conditions allow healthy prey, infected prey and predator to coexist? 3. n the presence of Allee effects, can disease save the population from predation-driven extinction? 4. n the presence of Allee effects, can predation save the population from disease-driven extinction? We will try to answer the questions above by ) obtaining a complete global picture of the population dynamics of the proposed susceptible prey-infected prey-predator interaction model S-Allee Model) as well as its corresponding model without Allee effects S-no-Allee Model); ) comparing the dynamics of the model with Allee effects to the one without Allee effects. The rest of the paper is organized as follows: n Section, we provide the detailed formulation of a general prey-predator system with prey subject to Allee effects and disease; and we show the basic dynamical properties of such general model. n Section 3, we obtain the complete dynamics of a concrete model when it is disease free and/or predation free i.e., the submodels of S-Allee Model); and we compare the dynamics to their corresponding models in the absence of Allee effects. n Section 4, we provide detailed analysis and its related numerical simulations to obtain the complete dynamical feature of this S-Allee model. Our results include sufficient conditions on its global attractors as well as its corresponding basins of attractions in different scenarios. n Section 5, we perform analysis of S-no- Allee Model under the same assumptions. n addition, we provide the biological implications on the impacts of Allee effects, disease and predation. n the last section, we conclude our findings and provide a potential future study Development of the model We start from the assumption that prey is facing an infectious disease that can be captured by an S Susceptible-nfected) framework where predator -class) feeds on both susceptible prey S-class) and infected prey -class). Let S be the normalized susceptible prey population;, denote the infected prey population and the predator population, respectively, both of which are relative to the susceptible prey population; and N = S + denotes the total population of prey. n the absence of disease and predation, we assume that the population dynamic of prey can be described by the following generic single species population model with an Allee effect: = rss θ) S) ) 6 6 where S denotes the normalized health prey population; the parameter r denotes the maximum birthrate of species, which can be scaled to be by altering the time scale; the parameter < θ < denotes

3 the Allee threshold normalized susceptible population). The population of ) converges to if initial conditions are below θ while it converges to if initial conditions are above θ. We assume that a) disease does not have vertical transmission but it is untreatable and causes an additional death rate; b) -class does not contribute to the reproduction of newborns; and c) the net reproduction rate of newborns is modified by the disease e.g, infectivies compete for resource but do not contribute to reproduction). Then in the presence of disease i.e., > ) and the absence of predation i.e., = ), the formulation of susceptible prey population dynamics can be described by the following ): = rss θ) S ) φn) }{{} N S }{{} the net reproduction modified by disease new infections ) where φn) is the disease transmission function that can be either density-dependent i.e., φn) = βn which is also referred to the law of mass action) or frequency-dependent i.e., φn) = β). Thus, the formulation of infective population can be described by the following 3), d = φn) N S }{{} µ. the natural mortality plus an additional mortality due to disease 3) n the presence of disease but in the absence of predation =, a general S model subject to Allee effects in prey can be represented as follows: = rss θ) S ) φn) N S d = φn) N S µ 4) where the parameter µ denotes the death rate of -class, which includes an additional disease-induced death rate. The S model 4) is a special case of an S model studied by Kang and Castillo-Chavez [48] where φn) = βn, ρ =, α = and α =. This modeling approach is similar to the work by Boukal and Berec [], Deredec and Courchamp [3], Courchamp et al. [8] and Hilker et al. [37] regarding the effects of Allee effects and disease our detailed approach of the host population without disease and predation is represented in Appendix). There are many literatures using this phenomenological model 4) to study the disease dynamics as well as invasion of pest e.g., see [54, 3,, 57, 36, 6, 7]). n the presence of predation, we assume that predator consumes S and -class at the rate of hs, N) and h, N), respectively, where -class has less or negative contribution to the growth rate of predator in comparison to S-class. The functional responses hs, N), h, N) can take the form of Holling-Type or or, i.e., Holling T ype : hs, N) = as; h, N) = a Holling T ype : hs, N) = as k+s+ ; h, N) = a k+s+. Holling T ype : hs, N) = as k +S+) ; h, N) = a k +S+) Therefore, a general predator-prey model where prey is subject to Allee effects and disease, is given by the following set of nonlinear differential equations: = rss θ) S ) φn) N S hs, N), d = φn) N S h, N) µ, 5) d = [chs, N) + γh, N) d]. 3

4 Allee Effects θ Susceptible opulation S) φn) N S N = S + Functional esponse nfected opulation ) µ Natural death and disease induced death c hs, N) γ h, N) redator opulation ) d Natural Death Figure : Schematic diagram of a general prey-predator model with prey subject to Allee effects and disease see the presentation of the model in 5)) where all parameters except γ are nonnegative. The parameter d represents the natural death rate of predator; the parameter c, ] is the conversion rate of susceptible prey biomass into predator biomass; and γ indicates that the effects of the consumption of infected prey on predator which could be positive or negative. More specifically, we assume that < γ < c; γ < indicates the consumption of infected prey increases the death rate of the predator see [6]), while γ > indicates the consumption of susceptible prey increases the growth rate of the predator. The biological significance of all parameters in Model 5) is provided in Table. The conceptual schematic diagram of this general model is presented in the schematic diagram. n summary, the formulation of a general S model 5) subject to Allee effects in prey is based on the following three assumptions: a) Disease does not have vertical transmission but it is untreatable and causes an additional death rate; b) Allee effects are built in the reproduction process of S-class which -class does not contribute to; c) redator consumes S and -class at the rate of hs, N) and h, N), respectively, whose growth rate is benefit less or even getting harm from -class. Our modeling assumptions are supported by many ecological situations. For example, in Salton Sea California), predatory birds get additional mortality though eating fish species that are infected by a vibrio class of bacteria and could also be subject to Allee effects see more discussions in [6, 5]). n nature, it is also possible that predator captures infected prey who is given up by predator due to its unpleasant taste or malnutrition from infections. We would like to point out that the assumption c) is critical to the dynamical outcomes of 5) as we should see from our analysis in the next few sections. To continue our study, let us define the state space of 5) as X = {S,, ) 3 +} whose interior is defined as X = {S,, ) 3 + : S > }. n the case that φn) = β, we define the state space as 4

5 6 7 X = {S,, ) 3 + : S + > }. Notice that hx, N) is chosen from Holling Type or or and φn) = βn or β, then the basic dynamical property of 5) can be summarized as the following theorem: Theorem. Basic dynamical features). Assume that c, ], d >, θ, ), < γ < c, µ > rθ. Then System 5) is positively invariant and uniformly ultimately bounded in X with the following property lim sup St) + t). 8 9 n addition, we have the following:. f φn) N µ, for all N >, then lim sup t) =.. f S) < θ, then lim max{st), t), t)} =. roof. For any S,,, we have =, d S= = and d = = = which implies that S =, = and = are invariant manifolds, respectively. Due to the continuity of the system, we can easily conclude that System 5) is positively invariant in 3 +. Choose any point S,, ) X such that S >, then due to the positive invariant property of 5), we have = rss θ) S ) φn) S hs, N) <. S> N n addition, since we have = and <, thus we can conclude that S=,=, = S=,+ > lim sup St). 3 Now we define the following two functions as Nt) = S + and Zt) = S + +, then we have dnt) dzt) = rss θ) N) µ [hs, N) + h, N)] rss θ) N) µ 6) = rss θ) N) µ d [hs, N) + h, N) chs, N) γh, N)]. 7) Since µ > rθ > rθ 4 and lim sup St), then for any ɛ >, there is a T large enough such that for any t > T, we have dnt) rss θ + µ/r) [rss θ) + µ] N r + ɛ) + ɛ θ + µ/r) ] [ rθ 4 + µ N. By applying the theory of differential inequality [] or Gronwalls inequality) and letting ɛ, we obtain lim sup Nt) = lim sup St) + t) r rθ + µ. µ rθ 4 This implies that both Nt) and t) are uniformly ultimately bounded. Similarly, since c, ] and < γ < c, then we have for any ɛ >, there is a T large enough such that for any t > T, dzt) = rss θ) N) µ d [hs, N) + h, N) chs, N) γh, N)] rss θ) N) µ d = L ɛ min{µ, d}z 5

6 where L ɛ = max { S +ɛ, N r rθ+µ µ rθ 4 +ɛ} { rss θ) N) + min{µ, d}s }. This implies that lim sup Zt) = lim sup St) + t) + t) L min{µ,d} where 4 5 L ɛ = max { S, N r rθ+µ µ rθ 4 Thus t) is also uniformly ultimately bounded. uniformly ultimately bounded in X. The fact that } { rss θ) N) + min{µ, d} }. Therefore, System 5) is positively invariant and = rss θ) S ) φn) N S hs, N) rs S ) S θ) φn) N S ) d = φn) N S h, N) µ φn) N S µ φn) N S µ implies that the dynamics of the S model 4) can govern the dynamics of S, -class in Model 5). f φn) N µ, then the S model 4) has no interior equilibrium since lim sup St). Then according to oincaré-bendixson Theorem [3], any trajectory of 4) converges to either a locally asymptotically stable equilibrium or a limit cycle. However, no interior equilibrium and no equilibrium on -axis indicates that any trajectory converges to a boundary equilibrium located on S-axis. Thus, we have lim sup t) = if φn) N µ. Assume that the initial susceptible prey population is less than θ and the initial infective population is large enough, the susceptible prey population can increase at the beginning due to the possibility of [ = rs) S) θ) S) )) φn)) ] ) >. t= N) However, the susceptible prey population can never increase to θ since [ = rs S θ) S ) φn) ] S=θ N = φn) S=θ N S S=θ <. This implies that Since φn) N St) < θ whenever S) < θ, for all t >. µ implies that lim sup t) =, thus the limiting dynamics is = rss θ) S) with St) < θ. This indicates the susceptible prey population will eventually converge to. Therefore, we have Now assume that φn) N S lim max{st), t), t)} =. > µ, for all N >. Since µ > rθ and lim sup St) θ, then we have [ ] rs θ) S ) φn) N < rs [S θ) S) S θ + µ/r)]. rs [S θ) S) θ + µ/r)] < rss θ) S) 6

7 This implies that lim St) =. Therefore, we have lim max{st), t), t)} = whenever S) < θ. 6 n the case that S) = θ, then we have St) < θ if ) + ) > or St) = θ if ) + ) =. Without loss of generality, let us assume S) + ) > and ) >. Then according to the argument above, we have lim max{st), t), t)} = lim sup St) + t) whenever there exists a T such that ST ) θ. Now assume that St) > θ, for all t, then we have dnt) = rss θ) N) µ [hs, N) + h, N)] rss θ) N) µ < whenever N) >. Therefore, we have lim sup Nt) = lim sup St) + t). 7 Notes: The assumption of µ > rθ follows from the fact that the natural mortality rate of the susceptible prey is rθ see the derivation of this assumption in the Appendix A). Theorem. indicates that our general prey-predator model with Allee effects and disease in prey has a compact global attractor living in the set { S,, ) X : S +, S + + max { N } { rss θ) N) + min{µ, d} } min{µ, d} }. 8 9 n addition, Theorem. implies that initial population of susceptible prey plays an important role in the persistence of S, or or due to Allee effects in prey. One direct application of Theorem. is presented as the following corollary: Corollary.. [ange of susceptible and infective population] Assume that c, ], d >, θ, ), < γ < c, µ > rθ. Then a necessary condition for the endemicity of the disease of System 5) is as follows: lim inf St) > θ and lim sup t) < θ. Theorem. and its corollary. provide the basic dynamical features of the general prey-predator model 5). n order to explore more complete dynamics of 5), we will focus on the case when φn) = βn and hx, N) = ax. Then, in the presence of both disease and predator, depending on whether infectives have a positive or negative impact on the growth rate of predator i.e., the sign of γ being positive or negative), the predator-prey model subject to Allee effects e.g., induced by mating limitations) and disease 5) can be written as the following if we scale away r i.e., r = ) : = SS θ) S ) βs as = S [S θ) S ) β a ] = Sf S,, ), d = βs a µ = [βs a µ] = f S,, ), 8) 7 8 d = a cs + γ) d = [bs + α d] = f 3 S,, ) where the parameter a indicates the attack rate of predator. For convenience, we let b = ac, a] and α = aγ, ac]. Variables and parameters used in Model 8) S-model) are presented in Table. 7

8 Variables/arameters Biological meaning S Density of susceptible prey Density of infected prey Density of predator θ Allee threshold β ate of infection a Attack rate of predator b The total effect to predator by consuming susceptible prey µ Death rate of infected prey c Conversion efficiency on susceptible prey γ Conversion efficiency on infected prey α The total effect to predator by consuming infected prey d Natural death rate of predator Table : Variables and parameters used Model 8) Notes: The term SS θ) S ) of in 8) models the net reproduction rate of newborns, a term that accounts for Allee effects due to mating limitations as well as reductions in fitness due to the competition for resource from infectives. Our model normalizes the susceptible population to be in a disease-free environment; and defines the infected prey population as well as the predator population relative to this normalization. Our modeling approach see the Appendix A) and assumptions a), b), c) require that the parameters of 8) are subject to the following condition: H: < θ <, µ > θ, < b = ac a and < α < b. The features outline above include factors not routinely considered in infectious-disease models. Allee effects are found in the epidemiological literature e.g., see [35, 7, 37]) as well as in the predator-prey interaction models [9, 75]. The rest of our article is focus on studying the dynamics of this simple S model 8) that incorporates Allee effects in its reproduction process, disease-induced additional death, and disease-induced effects on predation Dynamics of submodels n order to understand the full dynamics of 8), we should have a complete picture of the dynamics of the following two submodels:. The predator-prey model in the absence of the disease in 8) is represented as = S [S θ) S) a ] = Sf S,, ), d = [bs d] = f 3 S,, ). The submodel 9) has been introduced by other researchers e.g., [9, 74, 75]). For convenience, we introduce a disease-free demographic reproduction number for predator 9) = b d which gives the expected number of offspring b of an average individual predator in its lifetime 47 d. The reproduction number 48 is based upon the assumptions that the susceptible prey is at unit density i.e. S = ) and the disease is absent i.e. = ). The value of 49 < indicates that the predator cannot invade while the value of 5 > indicates that the predator may invade. 8

9 5. The S model in the absence of predation in 8) is represented as = S [S θ) S ) β] = Sf S,, ), d = [βs µ] = f S,, ). Kang and Castillo-Chavez [48] have studied a simple S model with strong Allee effects where they consider a susceptible-infectious model with the possibility that susceptible and infected individuals reproduce with the S-class being the best fit, and also infected individuals loose some ability to compete for resources at the cost imposed by the disease. The submodel ) is a special case of the S model studied by them where ρ =, α = and α =. We adopt the notations in Kang and Castillo-Chavez [48] and introduce the basic reproductive ratio ) = β µ whose numerator denotes the number of secondary infections βs = β per unit of time at the locally asymptotically stable equilibrium S = ) and denominator denotes the inverse of the average infectious period µ. The value of < indicates that the infection cannot invade while > indicates that the disease can invade. A direct application of Theorem. to the submodels 9) and ) gives the following corollary: Corollary 3. ositiveness and boundedness of submodels). Assume that both 9) and ) are subject to Condition H. Then both submodels are positively invariant and uniformly ultimately bounded in +. n addition, the submodel ) has the following property: lim sup St) + t) n the next two subsections, we explore the detailed dynamics of both submodels 9) and ). 3.. Equilibria and local stability t is easy to check that both submodels 9) and ) have, ), θ, ) and, ) as their boundary equilibria. For convenience, for Model 9), we denote E =, ), Eθ = θ, ), E =, ) and Ei =, ) a θ )) while for Model ), we denote E =, ), E θ = θ, ), E =, ) and E i =, ) θ θ + β where Ei 59, E i are interior equilibria for the submodel 9) and ), respectively, provided their existence. 6 The local stability of equilibria of both submodels 9) and ) can be summarized in the following 6 proposition: ) roposition 3.. [Local stability of equilibria for submodels 9) and )] The local stability of boundary equilibria of both submodels 9) and ) is summarized in Table while the local stability of interior equilibrium of both submodels 9) and ) is summarized in Table 3. Moreover, the equilibria Ei of the submodel 9) undergoes a supercritical Hopf-bifurcation at = θ+ and the equilibria E i of the submodel ) undergoes a supercritical Hopf-bifurcation at = β θ+ β βθ+β β+βθ θ. 9

10 Boundary Equilibria Stability Condition E and E Always locally asymptotically stable Eθ Saddle if < θ ; Source if > θ Eθ E Saddle if < θ ; Source if > θ Locally asymptotically stable if < ; Saddle if > E Locally asymptotically stable if < ; Saddle if > Table : The local stability of boundary equilibria for both submodels 9) and ) nterior Equilibrium Condition for existence Condition for local asymptotic stability E i < < θ < < θ+ E i < < θ < < β θ+ β βθ+β β+βθ θ. Table 3: The local stability of interior equilibrium for both submodels 9) and ) roof. The Jacobian matrix of the submodel 9) at its equilibrium S, ) is presented as follows [ ] J S S, ) = θ) S ) a + S S + θ) as b bs ) d while the Jacobian matrix of the submodel ) at its equilibrium S, ) is presented as follows [ ] J S S, ) = θ) S ) β + S S + θ) S S + θ β) β βs. ) µ After substituting S, ) = E u, u =, θ,, i into ), we obtain the eigenvalues for each equilibrium:. E =, ) is always locally asymptotically stable since both eigenvalues associated with ) at E are negative, i.e. λ = θ and λ = d.. Eθ = θ, ) is a saddle if < θ and is a source if > θ ) at Eθ can be represented as follows: since both eigenvalues associated with λ = θ θ) > ) λ = dθ ) { < if < θ θ > if > θ. 3. E =, ) is locally asymptotically stable if < and is a saddle if > since both eigenvalues associated with ) at E can be represented as follows: 7 7 λ = θ ) < ) λ = d ) { < if < > if >. 4. The unique interior equilibrium E i = S, ) = < < θ. The Jacobian matrix evaluated at E i is given by J E i = [ A B C ] = b a, a ) + θ ) ) θ ) )) θ a exists only if

11 whose characteristic equation is given by where BC > and This indicates that the eigenvalues of J E i λ Aλ + BC = A > if > + θ while A < if < + θ. are 7 or Therefore, E i λ = A A 4BC λ = A i 4BC A and λ = A + A 4BC and λ = A + i 4BC A exists and is locally asymptotically stable if Notice that A = when = +θ, and da d < < min{ θ, + θ } = + θ. ) = θ + ) θ + ) ) 3 with da d ) when A > 4BC when A < 4BC. = +θ = θ + )3 4 >, 73 thus according to Theorem 3..3 in Wiggins [77], we know that the submodel 9) undergoes a Hopf-bifurcation at = θ+. Then apply Theorem 3. from Wang et al. [75], we can conclude 74 that the Hopf-bifurcation is supercritical Similarly, after substituting S, ) = Eu, u =, θ,, i into ), we obtain the eigenvalues for each equilibrium:. E =, ) is always locally asymptotically stable since both eigenvalues associated with ) at E are negative, i.e. λ = θ and λ = µ.. Eθ = θ, ) is a saddle if < θ and is a source if > θ ) at Eθ can be represented as follows: since both eigenvalues associated with λ = θ θ) > ) λ = µθ ) { < if < θ θ > if > θ. 3. E =, ) is locally asymptotically stable if < and is a saddle if > since both eigenvalues associated with ) at E can be represented as follows: λ = θ ) < ) λ = µ ) { < if < > if >.

12 4. The unique interior equilibrium E i = S, ) = since from Condition H, we have, ) θ +β θ ) exists only if < < θ 78 + β θ = µ β + β θ > θ 4β + β θ = θ + 4β 4θβ θ β) =. 4β 4β The Jacobian matrix evaluated at Ei is given by J E i = [ A B C ] = β θ ) ) θ +β θ ) +β θ ) + θ + β θ) whose characteristic equation is given by where BC > and = A = ) ) θ + β θ λ Aλ + BC = ) + β θ) ) β θ) + β βθ + β β + βθ θ + θ = β + βθ θ )) β θ) ) +, β θ)) ) β θ) ) β βθ + β β + βθ θ. Thus, we have A > if > β θ + β βθ + β β + βθ θ while A < if < β θ + β βθ + β β + βθ θ. This indicates that the eigenvalues of J E i are or λ = A A 4BC and λ = A + A 4BC when A > 4BC λ = A i 4BC A and λ = A + i 4BC A when A < 4BC. Therefore, Ei exists and is locally asymptotically stable if < < min{ θ, β θ + β βθ + β β + βθ θ } = β θ + β βθ + β β + βθ θ Notice that A = when = β θ+ β βθ+β β+βθ θ and da d ) ββ θ) + β = = β θ+ β βθ+β β+βθ θ β + θ β θ)) ) + β θ) ) >. Thus according to Theorem 3..3 in Wiggins [77] and Theorem 3. in Wang et al. [75] again, we can conclude that the submodel ) undergoes a supercritical Hopf-bifurcation at = β θ+ β βθ+β β+βθ θ.

13 8 Notes: Local analysis results provided in roposition 3. and Table 3 suggest that the coexistence of prey and predation at the equilibrium Ei in the subsystem 9) is determined by the Allee threshold θ since Ei is locally asymptotically stable if < < θ + since θ + < θ. And the coexistence of health prey and infected prey at the equilibrium Ei in the subsystem ) is determined by both the Allee threshold θ and the disease transmission rate β since Ei is locally asymptotically stable if < < β θ + β βθ + β β + βθ θ since β θ + β βθ + β β + βθ θ < θ Disease/predation-driven extinctions and global features of submodels n this subsection, we focus on the disease/predation-driven extinctions as well as the features of global dynamics of both submodels. First, we have the following theorem regarding the extinction of one or both species: Theorem 3.. [Extinction] Assume that both submodels 9) and ) subject to Condition H. Then. f, then the population of predator in the submodel 9) goes extinction for any initial condition taken in + [see Figure a)]. Similarly, if, then the population of infectives in the submodel ) goes extinction for any initial condition taken in + [see Figure c)].. f θ, then System 9) converges to, ) for any initial condition taken in the interior of 9 +, which is predation-driven extinction [see Figure b)]. Similarly, if θ, then System ) 9 converges to, ) for any initial condition taken in the interior of 93 +, which is disease-driven 94 extinction [see Figure d)] f S) < θ, then all species in both submodels 9) and ) converge to, ). 96 roof. The detailed proof for the submodel 9) is similar to the proof for the submodel ), thus we 97 only focus on the submodel ). According to roposition 3., if or θ, then the submodel ) only has three boundary 98 equilibria Eu, u =, θ, where Eθ is a saddle and E is locally asymptotically stable when 99 < while Eθ is a source and E is a saddle when >. For =, E is nonhyperbolic with one zero eigenvalue and the other negative while Eθ remains saddle. For = θ, E θ is nonhyperbolic with one zero eigenvalue and the other positive while E remains saddle. 3 According to Theorem., the submodel ) has a compact global attractor. Thus, from an ap- 4 plication of the oincaré-bendixson theorem [3] we conclude that the trajectory starting at any initial condition living in the interior of + converges to one of three boundary equilibria Eu, 5 u =, θ, when ) has no interior equilibrium. This implies that lim sup t) = when or 6 θ. Since E is the only locally asymptotically stable boundary equilibrium when 7 θ, therefore, System ) converges to, ) for any initial condition taken in the interior of The third part of Theorem 3. can be a direct application of results from Theorem.. Therefore, the statement holds. 3

14 E E θ E E E θ E S S a) redation free for the submodel 9) b) redation-driven extinction for the submodel 9) E E θ E E E θ E S S c) Disease free for the submodel ) d) Disease-driven extinction for the submodel ) Figure : hase portraits of submodel 9) first row) and ) second row) when β =.6, θ =.4, a = and b =.. a)-e E is the global attractor when =.6 for the submodel 9); b)-e is the global attractor when = 3; c)-e E is the global attractor when =.5 for the submodel ); d)-e is the global attractor when =.6 for the submodel ). Notice that = b d and = β µ. 4

15 Notes: The second item in the statement of Theorem 3. is disease/predation-driven extinctions due to Allee effects of the susceptible population. The predation-driven extinction is also called overexploitation where both prey and predator go extinct dramatically due to large predator invasion [74, 75], i.e., predator reproduces fast enough to drive the prey population below its Allee threshold, thus lead to the extinction of both species. The biological explanation of disease-driven extinction is credited to the large disease transmission rate i.e., the basic reproduction number is large) while the reproduction of the susceptible population is not fast enough to sustain its own population. Thus, the susceptible population drops below its Allee threshold and decreases to zero, which eventually drives the infected population extinct eventually. The third item in the statement of Theorem 3. does not always hold if Condition H does not hold. For example, if we drop the assumption µ > θ, then the condition S) < θ does not always lead to the extinction of both susceptible and infective population in the submodel ) Global features of submodels 9) and ) The dynamics of global features of submodels 9) and ) are similar. Fix β =.6, θ =.4, a =, b =., and vary the basic reproduction numbers, for the submodel 9), the submodel ), respectively:. For the submodel 9): a) < : This leads to the predation free dynamics with E E as attractors according to Theorem 3. [see Figure a)]. b) < <.4857 = +θ : There is a transcriptical bifurcation at =. When increasing the value of from, E becomes unstable and the unique and locally asymptotically stable interior equilibrium Ei occurs is locally asymptotically stable [see roposition 3. and Figure 3b)]. c).4857 = +θ < < : There is a supercritical Hopf-bifurcation at =.4857 = +θ which leads to the unique stable limit cycle [see roposition 3. and Figure 35 3b)]. Wang et al. [75] has provided the proof of the uniqueness of the limit cycle. 36 d) At = : There is a heteroclinic bifurcation at 37 = [see Figure 3c)], i.e., there is a heteroclinic orbit connecting E to Eθ. The disappearance of the unique stable limit cycle is associated with the occurrence of heteroclinic connections: Outside the heteroclinic cycle the trajectory goes asymptotically to extinction equilibrium E 4, while for 4 initial conditions inside the heteroclinic cycle the trajectory converges towards the heteroclinic 4 cycle. Sieber and Hilker [63] and Wang et al. [75] have provided the proof of the existence of 43 the heteroclinic orbit. e) < < θ =.5: The predation-driven extinction occurs: the heteroclinic orbit 44 is broken and all trajectories in the interior of + converge to E 45 : For initial condition inside the curve bounded by the stable manifold of E 46, the orbit oscillates before finally converging slowly to E while all orbits above the unstable manifold of E converge towards E 47 [see 48 Figure 3d)]. f) 49 θ =.5: The predation-driven extinction occurs and the system has no interior equilibrium any more. All trajectories in the interior of + converge to E 5 [see Figure b)] For the submodel ): a) The submodel ) exhibits exactly the same dynamics feature as the submodel 9) when we increase the value of from : A transcritical bifurcation occurs at =, For < < β θ+ β βθ+β β+βθ θ =.54, the unique interior equilibrium E i is locally asymptotically stable [see Figure 3e)]. At = β θ+ β βθ+β β+βθ θ =.54, a supercritical Hopf-bifurcation occurs 5

16 E i... E i.8.8 E i.8 E i E E θ S E E E θ E E E θ E E E θ E S S S a) Stable interior equilibrium for the submodel 9) when =.4 b) Stable limit cycle for c) Heteroclinic orbit for the submodel 9) when the submodel 9) when =.43 = d) redation-driven extinction for the submodel 9) when = E i E i.. E i. E i E E θ E E E θ E E E θ E E E θ E S S S S e) Stable interior equilibrium for the submodel ) when =.5 f) Stable limit cycle for g) Heteroclinic orbit for the submodel ) when the submodel ) when =.548 = h) Disease-driven extinction for the submodel ) when = Figure 3: hase portraits of submodels 9) the first row) and ) the second row) when β =.6, θ =.4, a = and b =.. Notice that = b d and = β µ which leads to a unique stable limit cycle for.54 < < [see Figure 3f)]. The heteroclinic bifurcation occurs at = [see Figure 3g)] and disease-driven extinction occurs when > [see Figure 3h) and d)]. The impact of Allee effects: Without Allee effects, the submodels 9) and ) can be represented as the following two models: = S [ S a ], d = [bs d], d = S [ S β] = [βs µ]. 3) The two models above have the same dynamics as the traditional Lotka-Volterra edator-rey model: f k, k =,, then both models of 3) has global stability at, ); while if k >, k =,, then both models of 3) has global stability at its unique interior equilibrium. Compare this simple dynamics to the dynamics of submodels 9) and ), we can conclude that the effects of Allee effects:. mportance of initial conditions: Allee effects in the susceptible population, requires its initial condition being above the Allee threshold to persist.. Destabilizer: The nonlinearity induced by Allee effects destablizes the system which lead to fluctuated populations e.g., stable limit cycle). 3. Disease/predation-driven extinction: This occurs when the basic reproduction number of disease or predation is large enough to drive the susceptible population below its Allee threshold, thus all species go extinct. 6

17 7 4. Dynamics of the full S-- model After obtaining a complete dynamics of disease/predation free dynamics of the full S model 8) in the previous section, we continue to study the dynamics of the full model. We start with the boundary equilibria and their stability of 8). t is easy to check that System 8) has the following boundary equilibria: E =,, ), E θ = θ,, ), E =,, ), E i =,, ) a θ )) and E i =, ) ) θ + β θ, The existence of E i requires < < θ while the existence of Ei requires < < θ. roposition 4.. [Boundary equilibrium and stability] Sufficient conditions for the existence and the local stability of boundary equilibria for System 8) are summarized in Table 4. Boundary Equilibria Stability Condition E Always locally asymptotically stable E θ Source if > θ and > θ ; otherwise is saddle E Locally asymptotically stable if < and < E i Locally asymptotically stable if < < +θ and < + E i Locally asymptotically stable if < < β θ+ β βθ+β < α ) θ d +β θ) ) β+βθ θ Table 4: Sufficient conditions for the existence and local stability of boundary equilibria for System 8) ) ) θ µ and roof. The local stability of equilibrium can be determined by the eigenvalues λ i, i =,, 3 of the Jacobian matrix of System 8) evaluated at the equilibrium. By simple calculations, we have follows:. The equilibrium E =,, ) is always locally asymptotically stable since its eigenvalues are λ = θ < ), λ = µ < ), λ 3 = d < ). The equilibrium E θ = θ,, ) is always unstable since its eigenvalues are λ = θ θ) > ), λ = µθ ) { < if < θ, λ θ > if > 3 = dθ θ θ ) { < if < θ > if > θ 3. The equilibrium E =,, ) is locally asymptotically stable if < and < since its eigenvalues are λ = θ ) < ), λ = µ ) { < if < > if >, λ 3 = d ) { < if < > if > 7

18 where the sign of λ i indicates its eigenvector pointing toward < ) or away from > ) the equilibrium in S-axis i = ), -axis i = ) and -axis i = 3), respectively. ) )) According to roposition 3., the equilibrium E i =,, a θ is locally asymptotically stable if it is locally asymptotically stable in the submodel 9) and ) ) d = β ) E i θ ) µ < θ < + µ which indicates that disease is not able to invade at E i. ) Similarly, the equilibrium E i =, ) θ +β θ locally asymptotically stable in the submodel ) and ) ) d = bs + α d = b α θ E i + + β θ, is locally asymptotically stable if it is d < < ) ) α θ d + β θ) which indicates that predator is not able to invade at E i. 8 Therefore, we can conclude that E i is locally asymptotically stable if < < + θ and < + ) ) θ µ and E i is locally asymptotically stable if < < β θ + β βθ + β β + βθ θ and < ) ) α θ d. + β θ) 8 Notes: Notice that + θ < β θ + β βθ + β β + βθ θ < θ, thus according to roposition 4., both E i and Ei can be locally asymptotically stable if < < + θ, < < β θ + β βθ + β β + βθ θ and d d + β θ) α + β θ) ) θ ) < < + ) θ µ For convenience, let d = µ =, β =.5, θ =., then, according to Condition H, we have < = β µ =.5 < β θ + β βθ + β β + βθ θ.794, = b d = b α, + θ = , ). 8

19 and d +β θ) ) ).967 = d +β θ) α θ.95.33α, ) ) ) )) + θ µ = Thus, according to roposition 4., we have the following statement when d = µ =, β =.5, θ =.:. Both E i and Ei are locally asymptotically stable if the following inequalities hold [see the blue region of Figure 4a)] < b = <.67, 3b b +.b) b ).95.33α < b <, < α b E i is locally asymptotically stable and Ei is locally asymptotically stable in the S-plane but is unstable in 3 + if the following inequalities hold [see the green region of Figure 4a)] {.95.33α < b = 3b } <.67, b > max,.967 b, < α b. +.b) b ) 3. E i is locally asymptotically stable and Ei is locally asymptotically stable in the S -plane but is unstable in 3 + if the following inequalities hold [see the yellow region of Figure 4a)] {.95.33α < b = 3b } <.67, b < min,.967 b, < α b. +.b) b ) According to roposition 4., sufficient conditions for E i and Ei being locally asymptotically stable in the S -plane, S-plane, respectively, but being unstable in 3 + are as follows: and < + which is impossible when α since < < + θ, < < β θ + β βθ + β β + βθ θ ) ) θ < µ d d + β θ) α < + β θ) ) θ d d + β θ) α + β θ) ) θ ) when α. n addition, numerical simulations suggest that even if α, E i 84 and Ei cannot be locally asymptotically stable in the S -plane, S-plane, respectively, but being unstable in ) Global features n this subsection, we first explore sufficient conditions that lead to the extinction of at least one species of S,,. Our study gives the following theorem: Theorem 4.. [Basic global features]assume that System 8) is subject to Condition H. Then 9

20 Stability egions of Ei and Ei The existence of interior equilibria Ei.5 β=.5,µ=d=,θ=., =.5;=b.5 β=.5; µ=d=; =.5; =b.5.5 α α /+θ) =b i and a) Stability regions of the boundary equilibria E Ei b) egions of the existence of Ei Figure 4: Fix β =.5; µ = d = ; θ =. and a = 3. The left graph indicates the stability regions of the boundary i and E i : i) n the blue region, both equilibria are locally asymptotically stable; ii) n the green region, E i equilibria E i is locally asymptotically stable while E i is locally asymptotically stable while Ei is unstable; iii) n the yellow region, E is unstable. The blue region in the right graph is the region when System 8) has a unique interior equilibrium which is a saddle; while the white region in the right graph indicates no interior equilibrium.

21 . f, then lim t) =. f, in addition,, then lim max{t), t)} =. 9 While if and > θ, then lim St), t), t)) = E.. f α < and, then lim t) =. f, in addition,, then lim max{t), t)} =. f α <, and > θ, then for any initial condition taken in the interior of 3 +, we have While if α > and + α θ) d lim St), t), t)) = E., then lim t) =. f, in addition,, then lim max{t), t)} =. f α >, + α θ) d and > θ, then for any initial condition taken in the interior of 3 +, we have lim St), t), t)) = E All trajectories of System 8) converge to E if S) < θ. roof. f, then β = φn) N µ. According to Theorem., we have lim t) =, i.e., the limiting dynamics of System 8) is the submodel 9) which has only boundary equilibrium, ), θ, ) and, ) when. Then oincaré-bendixson Theorem [3] to 9), we can conclude that lim t) =. Therefore, we have lim max{t), t)} = if &. While if, in addition, we have > θ instead, then from Theorem 3. we can conclude that the omega 9 limit set of S -plane is E E θ E. Since 93 indicates that, for any ɛ >, all trajectories enter into the compact set [, B] [, ɛ] [, B] when time large enough, therefore, the condition 94 and > θ indicates that, for any ɛ >, all trajectories enter into the compact set M = [, ] [, ɛ] [, ɛ] when time large enough. Choose ɛ small enough, then the omega limit set of the interior of M is E 97 since E is locally asymptotically stable and E θ, E is unstable according to roposition 3.. Therefore, the condition and > θ indicates that lim 98 St), t), t)) = E. f α >, then from the proof of Theorem 3. and Corollary., we can conclude that lim sup t) < θ. This indicates that for any ɛ >, there exists a time T such that ) d < b + ɛ) + α θ + ɛ) d = d α θ) ɛb + α) + + for all t > T d d which implies that lim t) = if + α θ) d <. f + α θ) d =, then we can apply oincaré- Bendixson Theorem [3] to ) to obtain that lim t) =. The rest of the second item of Theorem 4. can be shown by applying the similar arguments of the proof for the first item in Theorem 4.. The third item of Theorem 4. can be shown by a direct application of Theorem., i.e., all trajectories converge to E whenever S) < θ. Notes: A direct implication of Theorem 4. is that the coexistence of S,, population in System 8 requires > and > when α < ; α θ) > when α >. d

22 One interesting question is that if α > but α θ) d <, then what happens to the dynamics of System 8, e.g., can predator be able to persist under certain conditions? This has been partially answered by Theorem 4.: System 8 has no interior equilibrium as long as. n fact, predator is not able to survive in this case. 4.. The interior equilibrium f System 8) has a locally stable interior equilibrium, then we can say that S,, -class can coexist under certain conditions. Thus, in this subsection, we explore sufficient conditions for the existence of the interior equilibrium and its stability for System 8). For convenience, let where B = β θ) + S = B B 4C d α b α, C = µ θ b f β > µ, i.e., >, then we have follows B 4C = dβ θ) α α and E i k = S k, k, k ), k =,, S = B + B 4C, k = β Sk ) a, k = b α ) d α b α β θ) 4αµ θ) 4dβ θ) b α = β θ) + d α) b α) + 4dβ θ) 4αµ θ) d α)β θ) b α { <β θ) + d α) b α) + 4dβ θ) 4αβ θ) d α)β θ) b α S k = d α b α β θ)) if α< ), k =,.. >β θ) + d α) b α) + 4dβ θ) 4αβ θ) d α)β θ) b α = d α b α β θ)) if α> 39 Therefore, we can conclude that when >, we have S < d α b α if α < S > d α b α if α >. 4) n the case that µ = β i.e., = ), we have S = d α b α and S <. Now we have the following theorem regarding the number of interior equilibrium and its local stability: Theorem 4.. [nterior equilibrium]assume that Condition H holds for System 8).. System 8) has no interior equilibrium if one of the following conditions is satisfied: a) { } or {, α < b } or b) { } { α >, θ or α >, > } or { c) α <, = d b < d α b α < max { } } θ, or d) { α > } { d µ θ β θ or µ < θ + b α) θ β+ d α b α ) + 4dβ θ) b α 4α ) }. n the case that α >, > θ and > θ, every trajectory of System 8) with an initial condition taking in the interior of 3 + converges to E, i.e., lim St), t), t)) = E.

23 . System 8) has at most one interior equilibrium E i = S,, ). The existence of E i requires provided that max{θ, } < S < min{, α < dβ θ) µ θ = d µ θ β θ } when α >, or, > S > max { θ,, } when α < n addition, the real parts of all eigenvalues of the Jacobian Matrix evaluated at E i can never be all negative. roof. Direct applications of Theorem 4. imply that System 8) has no interior equilibrium if < or α <, < ) or α >, + α θ) d Thus, we omit the detailed proof for these cases. f S,, ) is an interior equilibrium for System 8), then S is a positive root of the quadratic equation S θ) S b )) α S β b ) α S β S ) = S BS + C = 5) < ). 33 provided that and = β a B = β θ) + d α b α, C = µ θ b S ) >, = b α α dβ θ) α S ) >. 6) The equation 6) implies that a necessary condition for the existence of the interior equilibrium S,, ) is as follows: < S < if α > ; while S > max {, } if α <. n the case that = µ β =, we have S = d α b α, thus the interior equilibrium S,, ) exists if < S < if α > ; while S > max {, } if α <. This is a contradiction to lim sup St) according to Theorem.. This implies that there is no interior equilibrium if =. Notice that Theorem 4. indicates that one necessary condition for System 8) having an interior equilibrium is that otherwise lim t) =, thus, there is no interior equilibrium if. ecall that Theorem. and Theorem 4. indicate that θ < S <. Therefore, the existence of an interior equilibrium S,, ) requires > i.e., µ < β) and max{θ, } < S < min{, } if α > ; while > S > max { θ,, } if α <. 3

24 This implies that there is no interior equilibrium S,, ) if or max{θ, max { θ, Therefore, there is no interior equilibrium if } > min{,, } when α > } when α <., α > ) or θ, α > ) or, α < ). Since we assume that System 8) satisfies Condition H, thus we have < θ <, µ > θ, < b a and < α < b. The requirement > implies that θ < µ < β. The equation 5) has only one positive root S = B+ B 4C if C = µ θ dβ θ) α αµ θ) dβ θ) dβ θ) b α = < < α < b α µ θ = d. µ θ β θ Therefore, System 8) has a unique interior equilibrium E i = S,, ) where S = B+ B 4C if α < d, max{θ, µ θ β θ } < S < min{, } when α >, or, > S > max { θ,, } when α <. n the case that α <, it is easy to check that C < since θ < µ < β implies that α < whenever α <. Thus, it is impossible that 5) has two positive roots when α <. f 5) has two positive roots S = B B 4C < S = B + B 4C, then it requires that α > and B > β θ) + d α b α > < β θ < d α b α C > αµ θ) dβ θ) > < β θ < αµ θ) b α B > 4C β θ) + Thus, B > and C > require that d α b α ) + β θ)d+α) 4αµ θ) b α >. d, s.t. α < min{b, d}, d µ θ β θ holds < β θ < d α b α s.t. < α < min{b, d} and < β θ < αµ θ) d < µ θ which is a contradiction since < µ θ < β θ and b > α. Therefore, System 8) has at most one interior equilibrium E i and System 8) has no interior equilibrium if C > or B < 4C which implies follows:. C > α > d µ θ. β θ. B 4C < µ < θ + b α) d α θ β+ b α) + 4dβ θ) b α 4α ) 4