Complex dynamical behaviour of Disease Spread in a Plant- Herbivore System with Allee Effect

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 6 Ver. III (Nov. - Dec. 5), PP Complex dnamical behaviour of Disease Spread in a Plant- Herbivore Sstem with Allee Effect Vijaalashmi.S, Gunasearan.M, Department of Mathematics, D.R.B.C.C.C. Hindu College, Chennai 67 and Research scholar, PG Department of Mathematics, Sri Subramania Swami Government Arts College, Tiruttani-639 PG Department of Mathematics, Sri Subramania Swami Government Arts College, Tiruttani-639 Abstract: This paper discusses the complex dnamical behaviour of a communicable disease in a plantherbivore sstem with Allee effect. It is assumed that (a) Disease has no vertical transmission but it is untreatable and causes additional mortalit in infected plant; (b) Allee effects built in the reproduction of susceptible plant while infected plant has no reproduction; (c) Herbivore captures susceptible and infected plant at the same rate but the consumption of infected plant has less benefits or even causes harm to herbivore; (d) Disease transmission follows the law of mass action; (e) Two predation response functions of Holling-Tpe II are used for both health and infected plants. The feasibilit and stabilit conditions of the equilibrium points of the sstem were analed.finall we performed numerical simulations to verif the theoretical results and to investigate further the properties of the sstem. Kewords: Allee effect, Holling-Tpe II, susceptible plant -infective plant, vertical transmission, law of mass action, stabilit analsis. I. Introduction Plant-Herbivore interaction is one of basic interspecies relations for ecological and social models. The first differential equation model of predator-pre tpe, called Lota-Volterra equation, was formulated b Alfred Lota and Vito Volterra in 9 s, when attempts were first made to find ecological laws of nature [6, ]. These equations expressed the relationship between two or more species. Man modifications and extensions have been made on the equations. One of them involves dividing one or two species into Susceptible and Infected class. Mathematical modeling has been a great tool for understanding species interactions as well as the disease dnamics, which allow us to obtain useful biological insights and enable us to mae correct policies to maintain the diversit in nature [7, ]. Eco-epidemiolog is comparativel a new branch in mathematical biolog which simultaneousl considers the ecological and epidemiological processes [3]. Hadeler and Freedman first introduced an eco-epidemiological model regarding predator-pre interactions with both pre and predator subject to disease [, 5]. Man mathematical models have been used to understand the impacts of Allee effects on species abundance and persistence especiall in the presence of disease [4-7]. Allee effects have been recentl studied intensivel in population dnamics. The Allee effect phenomenon was introduced b Warden Clde Allee in 93 and was widel publicied b the recent boo b Courchamp et al [, 4].Allee proposed that the per capita birth rate declines at low population densities. At the time, Allee s ideas were revolutionar since most of the biolog literature was advocating the negative densit dependence principle where higher population densit would limit the population growth. Empirical evidence of Allee effects has been reported in man natural populations including plants, insects, marine invertebrates, birds and mammals. Various mechanisms at low population sies, such as the need of a minimal group sie necessar to successfull raise offspring, produce seeds, forage, and/or sustain herbivore attacs or enhanced genetic inbreeding have been proposed as potential sources of the Allee effect. Former studies have demonstrated that Allee effects can have important dnamical effects on the local stabilit analsis of population models [8-]. It ma have either a destabiliing role or a stabiliing role in the sstem. The local stabilit of an equilibrium point ma be changed from stable to unstable or vice versa. It is also possible that for a population subject to an Allee effect, the sstem ma tae a much longer time to reach its stead state even though it is stable at an equilibrium point. The Allee effect has been attracting much attention recentl owing to its strong potential impact on population dnamics. It is widel accepted that the Allee effect ma increase the extinction ris of low-densit populations. Therefore the population ecolog investigation of the Allee effect is important to conservation biolog. In this paper, we propose a general plant-herbivore model with plant subject to Allee effects and disease. There are three unique features of our assumptions: (a) Disease has no vertical transmission but it is untreatable and causes additional mortalit in infected plant; DOI:.979/ Page

2 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect (b) Allee effects built in the reproduction of susceptible plant while infected plant has no reproduction; (c) Herbivore captures susceptible and infected plant at the same rate but the consumption of infected plant has less benefits or even causes harm to herbivore. These assumptions contribute great impacts on the dnamical outcomes of the proposed model. To explore how interpla among Allee effect, disease and predation affect species abundance and persistence, we focus on a concrete sstem with additional two assumptions: (d) Disease transmission follows the law of mass action; (e) Plant and Herbivore have Holling-Tpe II functional responses. II. Mathematical Model We illustrate a general plant-herbivore model where plant is subject to Allee effect and disease is given b the following extended model in Figure. Allee effect S.Plant (x) N = x + Functional response mh(x,n) F N x N I.Plant () μ ωh(,n) Natural death& disease induced Death Herbivore() Natural death Figure : Plant-Herbivore Model where Plant is subject to Allee Effect and Disease The extended sstem of Ordinar Differential Equations is dx dt = rx x x F N x H(x, N) N d dt = F N x H, N μ N () d = [mh x, N + ωh, N ] dt The parameters were described in Table Variables/Parameters Description x Densit of Susceptible Plant Densit of Infected Plant Densit of Herbivore N = x + Total population of Plant r Maximum birth-rate of species, which can be scaled to be b altering the time scale Allee threshold; < < μ Death rate of infected plant; μ > m Conversion efficienc on susceptible plant; < m Natural death rate of herbivore F N Disease transmission function that can be either densit-dependent (that isf N = N which is also referred to the law of mass action) or frequenc-dependent(that isf N = ) Rate of infection ω Conversion efficienc on infected plant; < ω < m ω < Consumption of infected plant increases the death rate of the herbivore ω > Consumption of susceptible plant increases the growth rate of the herbivore H x, N = ax ; +x+ H, N = a The functional responses can tae the form of Holling-Tpe II a b = am α = aω +x+ x x x Half-saturation constant Attac rate of herbivore The total effect to herbivore b consuming susceptible plant; < b a The total effect to herbivore b consuming infected plant; < α b 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 infective Table : Variables and Parameters used Model () and () DOI:.979/ Page

3 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect In the presence of predation, we assume that herbivore consume susceptible and infected plants at the rate of H x, N = ax and H, N = a respectivel, where infected plant has less or negative contribution +x+ +x+ to the growth rate of herbivore in comparison to susceptible plant. Based on the assumptions, the mathematical model is formulated as follows. dx dt = x x x x ax + x + d dt = x a + x + μ d dt = bx + x + + α + x + () III. Dnamics of submodels In order to understand the full dnamics of sstem (), we should have a complete picture of the dnamics of the following two submodels: 3. Sub-model: I The plant-herbivore model in the absence of the disease in () is represented as dx dt = x x x ax + x d dt = bx + x (3) For convenience, we introduce a disease-free demographic reproduction number for herbivorer = b which gives the expected number of offspring b of an average individual herbivore in its lifetime. The reproduction number is based upon the assumptions that the susceptible plant is at unit densit and the disease is absent. The value of R < indicates that the herbivore cannot invade while the value of R > indicates that the herbivore ma invade. Equilibrium points The sstem (3) has four equilibrium points which are i) E =, ii) E =, iii) E 3 =, iv) E 4 = (x, ) where x = and = a + Dnamical behaviour In this section, we stud the local behaviour of the sstem (3) about each equilibrium points. The stabilit of the sstem (3) is carried out b computing the Jacobian matrix corresponding to each equilibrium point. The Jacobian matrix is given b x x + x x + a ax J ( + x) x, = + x (4) b bx ( + x) + x Proposition: The equilibrium pointe =, is locall asmptoticall stable if Det (J, ) > and Tr (J, ) <. The Jacobian matrix evaluated at E is given b J, = The eigenvalues of J, are DOI:.979/ Page

4 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect = ; =. Beltrami conditions that, ifdet J > and Tr J <,then the equilibrium state of the model is locall asmptoticall stable [3]. Here, Det (J, ) = > and Tr (J, ) = ( ) <. Hence the conditions of Beltrami were satisfied. Therefore, we concluded that the equilibrium state E =, is locall asmptoticall stable. Proposition: The equilibrium pointe =, is a (a) Saddle if < + (b) Source if > + The Jacobian matrix evaluated at E is given b The eigenvalues of J, are = > & = b + J, = = + Therefore, E =, is a saddle if < + ( ) a + b + + = < if < + > ifr > + and is a source if R > + Proposition: 3 The equilibrium pointe 3 =, is a (a) Stable if R < ( + ) (b) Saddle if R > ( + ) The Jacobian matrix evaluated at E 3 is given b a J +, = b + The eigenvalues of J, are = < ; = b = R = < if < ( + ) > ifr > ( + ) Therefore, E 3 =, is locall asmptoticall stable if R < + and is a saddle if R > ( + ) Proposition: 4 The equilibrium point E 4 exists and is locall asmptoticall stable if + < < min +, + The unique interior equilibrium ( + ) < < + J x, = E 4 = x, =, a.the Jacobian matrix evaluated at E 4 is given b = A B C R R + R ( b a ( ) R + ) + a ( + ) Whose characteristic equation is given b A + BC = where BC > and + exists onl if A < if < + A > if > DOI:.979/ Page

5 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect This indicates that the eigenvalues of J x, are = A A 4BC (or) = A i 4BC A and = A+i 4BC A asmptoticall stable if + < < min +, + and = A+ A 4BC when A > 4BC when A < 4BC. Therefore, E 4 = (x, ) exists and is locall + = Sub-model: II The plant-herbivore model in the absence of predation in () is represented as dx = x x x x dt d = x μ dt (5) We introduce the basic reproductive ratior = whose numerator denotes the number of secondar μ infections per unit of time and denominator denotes the inverse of the average infectious period μ. The value of R < indicates that the infection cannot invade while the value of R > indicates that the disease can invade. Equilibrium points The sstem (5) has four equilibrium points which are i) E 5 =, ii) E 6 =, iii) E 7 =, iv) E 8 = (x, ) wherex = and = + Dnamical behaviour In this section, we stud the local behaviour of the sstem (5) about each equilibrium points. It is carried out b computing the Jacobian matrix corresponding to each equilibrium point. The Jacobian matrix is given b J x x + x x x(x ) x x x x, = (6) x μ Proposition: 5 The equilibrium pointe 5 =, is locall asmptoticall stable ifdet (J, ) > andtr (J, ) <. The Jacobian matrix evaluated at E 5 is given b J, = μ The eigenvalues of J, are = and = μ. Beltrami conditions that, if the Det (J) > and Tr J < then the equilibrium state of the model is locall asmptoticall stable. Here, Det (J, ) = μ > and Tr (J, ) = μ <. Hence the conditions of Beltrami were satisfied. Therefore, we concluded that the equilibrium state E 5 =, is locall asmptoticall stable. Proposition: 6 The equilibrium pointe 6 =, is a (c) Saddle if < (d) Source if > The Jacobian matrix evaluated at E 6 is given b J, = ( ) μ DOI:.979/ Page

6 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect The eigenvalues of J, are = > and = μ = μ = < ifr < > ifr > Therefore, E 6 =, is a saddle if R < and is a source if R > Proposition: 7 The equilibrium pointe 7 =, is a (c) Stable if < (d) Saddle if > The Jacobian matrix evaluated at E 7 is given b J, = μ The eigenvalues of J, are = < and = μ = μ R = < ifr < > ifr > Therefore, E 7 =, is locall asmptoticall stable if R < and is a saddle if R >. Proposition: 8 The equilibrium point E 8 exists and is locall asmptoticall stable if < < min, The unique interior equilibrium E 8 = (x, ) =, + exists onl if < <. Sincethe condition < <, μ >, < b a and < α b, We have, + = μ + > + =. The Jacobian matrix evaluated at E is given b J x, = ( ( + ) A B = C Whose characteristic equation is given b A + BC =, where BC > and = A = ( + + R ( ) + (R )(R + + ) Thus, we have A > if R > + + and A< if R + < This indicates that the eigenvalues of J x, are = A A 4BC = A i 4BC A and = A+i 4BC A asmptoticall stable if < < min, when A < 4BC. Therefore, E 8 = and = A+ A 4BC. when A > 4BC (or) = ( x, ) exists and is locall DOI:.979/ Page

7 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect IV. Disease/predation-driven extinctions of submodels In this section, we focus on the disease/predation-driven extinctions of both submodels. First, we have the following theorem regarding the extinction of one or both species: Theorem: If the submodels I and II with condition that < <, μ >, < b a, < α < bthen (a) If ( + ), then the population of herbivore in the submodel (3) goes extinction for an initial condition taes in +. (b) If, then the population of infective in the submodel (5) goes extinction for an initial condition taen in + (c) IfR +, then sstem (3) converges to (, ) for an initial condition taen in the interior of +, which is predation-driven extinction. (d) IfR, then sstem (5) converges to (, ) for an initial condition taen in the interior of +, which is disease - driven extinction. (e) If x <, then all species in both submodels (3) and (5) converge to (, ). Case: If ( + ) or + (,,),(,,),(,,), where E is a saddle when < +, then the submodel I onl has three boundar equilibrium and E is locall asmptoticall stable when R < ( + ) while E is a source whenr > + and E is a saddle when R > ( + ). For R = ( + ), E is non-hperbolic with one ero eigenvalues and the other negative while E remains saddle. ForR = +, E is non-hperbolic with one ero eigenvalues and the other positive while E remains saddle. Case: If R orr, then the submodel II onl has three boundar equilibrium (,,),(,,),(,,), where E is a saddle when R < and E is locall asmptoticall stable when R < while E is a source when R > and E is a saddle when R >. For R =, E is non-hperbolic with one ero eigenvalues and the other negative while E remains saddle. For R = +, E is non-hperbolic with one ero eigenvalues and the other positive while E remains saddle. V. Dnamics of a general plant-herbivore model where plant is subject to Allee effects and disease In this section, we stud the local behaviour of the sstem () about each equilibrium points. The stabilit of the sstem () is carried out b computing the Jacobian matrix corresponding to each equilibrium point. The Jacobian matrix is given b J x,, = a (+) x x + x x x(x ) x x x + ax (+x+) (+x+) a (+) x μ (++) b(+) α bx α (+x) + (+x+) (+x+) (+x+) (+x+) ax +x+ a (+) (++) bx + α +x+ +x+ -- (7) Proposition: 9 The equilibrium pointe 9 =,, is locall asmptoticall stable. The Jacobian matrix evaluated at E 9 is given b J,, = μ The eigenvalues of J,, are =, = μand 3 =. Itis alwas locall asmptoticall stable since its eigenvalues are negative. Proposition: The equilibrium pointe =,, is unstable. The Jacobian matrix evaluated at E is given b DOI:.979/ Page

8 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect J,, = ( ) a + μ b + The equilibrium pointe =,, is alwas unstable since the eigenvalues are = >, = μ= μ = < ifr < > ifr > and 3 = b = = < if < + > ifr > + Proposition: The equilibrium pointe =,, is locall asmptoticall stable if R < + and R <. The Jacobian matrix evaluated at E is given b J,, = a + μ b + The equilibrium point E =,, is locall asmptoticall stable if R < + and R < since the eigenvalues of J,, are = <, =( μ) = μ(r ) = < ifr < > ifr > and 3 = b = (R + + ( + )) = < if < ( + ) > ifr > ( + ) Proposition: The equilibrium point E = ( x,, ) = asmptoticall stable. The Jacobian matrix evaluated at E is given b J x,, = x x + x x x x The equilibrium point E = ( x,, ) =,, a a x x x + ax + x + x x μ a + b bx + x + x + α + x,, a asmptoticall stable if it is locall asmptoticall stable in the submodel I and R + = R R R R μ + that disease is not able to invade at E. < if and onl if < + d + ax + x bx + x + = x μ a dt R R + R μ + (8) is locall is locall +.At E, Therefore, we can conclude that the equilibrium point E =(x,, ) is locall asmptoticall stable if < + R R + R μ + and + < < + +. d dt which indicates Proposition: 3 The equilibrium point E 3 =(x,, ) =, The Jacobian matrix evaluated at E 3 is given b +, is locall asmptoticall stable. DOI:.979/ Page

9 Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect J x,, = ax x x + x x x(x ) x x x + x + x μ a ( + ) bx + x + + α + x + The equilibrium pointe 3 =(x,, ) = asmptoticall stable in the submodel II and, At the equilibrium point E = (x,, ), < α + d = bx + dt +x+ d dt = DOI:.979/ Page _(9), is locall asmptoticall stable if it is locall α +x+. R + + α + + R + R R + < if and onl if which indicates that herbivore is not able to invade at E 3.Therefore, we can conclude that the equilibrium point E =(x,, ) is locall asmptoticall stable if < α + and < < VI. Numerical Analsis In this section, we illustrate the analtical findings with the help of numerical simulations performed with MATLAB programming. We present time series plots for sstem (3) and (5) to confirm the theoretical results with hpothetical set of data. The show interesting complex dnamical behaviours. Figure: (See Appendix) shows the time series graph for Susceptible Plant and Herbivore with Allee effect when =.3, a = 6, = 6.8, b = 4.5 and =.6 Figure: (See Appendix) shows the time series graph for Susceptible Plant and Infected Plant with Allee effect when =.3, =.5 and μ = VII. Conclusion A mathematical model describing the spread of a disease in a plant-herbivore sstem with Allee effect has been discussed. An infectious disease is assumed to spread onl among plants and herbivore consumes both health and infected plants with two different predation responses. In the presence of Allee effects, species are prone to extinction and initial condition plas an important role on the surviving of plant as well as its corresponding herbivore and in the presence of Allee effects, disease ma be able to save plant from the predation-driven extinction and leads to the coexistence of susceptible and infected plant population while herbivore cannot save the disease-driven extinction. In this investigation, the stabilit of different equilibrium points has been thoroughl examined. Numerical simulations were done to observe the effect of the disease and Allee effect on the species and diagrams were presented which are supporting our results. All these findings ma have potential applications in conservation biolog. References [] Allee, W.C., 93. Animal Aggregations: A stud in General Sociolog. Universit of Chicago Press, Chicago. [] Asrul Sani, Edi Cahono, Muhsar, Gusti Arviana Rahman, Yuni Tri Hewindati, Faeldog, Farah Aini Abdullah, 4. Dnamics of Disease spread in a predator-pre sstem. Advanced Studies in Biolog, Volume -6 and Number 4, [3] Beltrami, E., 989. Mathematics for dnamic modeling, Academic Press, New Yor [4] Courchamp, F., Berec and Gascoigne, J., 8. Allee effects in Ecolog and Conservation. Oxford Universit Press, Oxford. [5] Hadeler, K.P., Freedman, H.I., 989. Predator - pre populations with parasitic infection. Journal of Mathematical Biolog, 7, [6] Lota, A.J., 95. Elements of phsical biolog, Williams and Wilins (Baltimore) [7] Md. Sabiar Rahman, Santabrata Charavart, 3. A Predator - Pre model with disease in pre. Nonlinear Analsis: Modeling and Control, Volume 8, Number, 9 9. [8] Merdan, H., Karaoglu, E.,. Consequences of Allee effects on stabilit analsis of the population model X t+ = X t f X t 3 Hacettepe Journal of Mathematics and Statistics, Volume 4(5), [9] Sujatha, K., Gunasearan, M., 5. Two was stabilit analsis of predator-pre sstem with diseased pre population, IJES

10 SUSCEPTIBLE PLANT & HERBIVORE SUSCEPTIBLE PLANT & INFECTED PLANT Complex dnamical behaviour of Disease Spread in a Plant-Herbivore Sstem with Allee Effect [] Vijaalashmi, S., Gunasearan, M., 5. Dnamical analsis of a discrete-time plant-larva-adult model with Allee effect on the plant, IJSR, Volume 4, Issue 7, [] Volterra, V., 96. Fluctuations in the abundance of species, considered mathematicall, Nature [] Wuhaib, S.A., AbuHasan, A., 3. A predator-infected pre model with harvesting of infected pre. Research Article, Science Asia39S,37 4. [3] YunKang, Carlos Castillo-Chave, 4. A simple epidemiological model for populations in the wild with Allee effects and Disease - Modified fitness, Discrete and continuous dnamical sstems series B, Volume 9, Number, 89 3 [4] YunKang, Carlos Castillo-Chave, 4. Dnamics of SI models with both horiontal and vertical transmissions as well as Allee Effects, Mathematical Biosciences, 48, [5] YunKang, Sourav Kumar Sasmal, Amia Ranjan Bhowmic and Jodev Chattopadha, 4. Dnamics of a Predator-Pre sstem With Pre subject to Allee effects and disease. Mathematical Biosciences and Engineering, Volume, Number 4. [6] YunKang, Sourav Kumar Sasmal, Amia Ranjan Bhowmic and Jodev Chattopadha, 4. A Host-Parasitoid sstem wit Predation-Driven component Allee effects in host population. Journal of biological dnamics [7] YunKang and Oita Udiani, 4. Dnamics of a single species Evolutionar model with Allee effects Appendix: HERBIVORE.6.4 SUSCEPTIBLE PLANT SUSCEPTIBLE PLANT. INFECTED PLANT TIME Figure: Time series graph for susceptible plant and Herbivore with Allee Effect ( =.3, a = 6, = 6.8, b = 4.5, =.6) TIME Figure: Time series graph for susceptible plant and Infected plant with Allee Effect ( =.3, =.5, μ = ) DOI:.979/ Page