Existence, Uniqueness Solution of a Modified. Predator-Prey Model

Size: px
Start display at page:

Download "Existence, Uniqueness Solution of a Modified. Predator-Prey Model"

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 4, 6, no. 4, HIKARI Ltd, Existence, Uniqueness Solution of a Modified Predator-Prey Model M. A. Al Qudah Mathematical Science Department Princess Nourah Bint Abdulrahman University P.O. Box 8448, Riyadh 67, Saudi Arabia Copyright 6 M. A. Al Qudah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A modified predator-prey model which describes the interaction between the Predator-Prey species and at the species itself is considered. The existence and uniqueness of solutions of a modified predator-prey model in L p,q space is proved under certain conditions of p and q. Some properties of the dynamically system of such model are described such as its stability. Graphs of solutions displayed. Applications of the results in mathematical biology are discussed. Mathematics Subject Classification: 9B99 Keywords: Predator-Prey model, existence, uniqueness, Stability. Introduction The dynamics of population has been described using mathematical models which have been very successfully for studying animals and human populations. Lotka [] initiated the predator-prey models and competing species relations. Levin and Segel [] studied some biological hypotheses concerning the origin of Planktonic Patchiness model. Fife [] considered reaction and diffusion systems which are distributed in 3-dimentional space or on a surface rather than on the line. Calderon [9] studied the diffusion and non-linear population theory. Abualrub [] discussed diffusion problems in mathematical biology. In addition, Abualrub [] studied diffusion in - dimensional spaces for which diffusion is more realistic and applicable in life and he proved the existence and uniqueness of long range diffusion reaction model on population dynamics. Abualrub [4] proved

2 67 M. A. Al Qudah the existence and uniqueness of solutions of a diffusive predator-prey model. Al- Qudah and Abualrub [5] studied the existence and uniqueness of long range diffusion involving flux for insect dispersal model, and they studied in [7] solitary and traveling wave solutions with stability analysis also for insect dispersal model. In this paper, we make another modification of the model of Abualrub [3] which is a modification to planktonic patchiness model, - a kind of predator-prey model-, for more details see []. Existence and uniqueness of solutions for the modified model will be considered. A traveling wave solutions and stability have been considered. This paper is organized as follows. In Section, a modified Predator- Prey model is considered. The existence and uniqueness of its solutions is given in Section 3. The stability analysis and the conclusion are given in Section 4.. The predator- prey model The model we are going to consider here is another modification to the planktonic patchiness model, a kind of predator-prey models, which were originally considered by Levin and Segel in 976 []. We use the model in [4, 3, 6] with the assumption that the species specific diffusion coefficients be constant to come up with the predator- prey model: u t u = a u + a u a 3 uv, () v t v = a 4 uv a 5 v. () Where u = u(x, y) is the prey density population, v = v(x, y)is the predator density population, x = (x, x ), and = x + x represents the diffusion (dispersal), we shall assume that a,,a 5 are constants and a may assumed to be compact supported and bounded function of x; that is ( a = a (x); a (x) = if x > N; where N; is a constant ) not a constant this is assumed, because the birth (or death) rate may depend on the environment, which is assumed to be bounded. Another reason for assuming that a,,a 5 are constants, is due to the fact that the birth(or death) rate depends on the interaction between the male and the female (sexual interaction as in terms a u and a 5 v ) or on the binary interaction between the males and females of the prey and the predator respectively ( as in the term uv). In this paper we assume that a is constant and we modify the model by assuming another kind of interaction between the prey and the predator, that is two terms are added, namely to get the following model : u t D u = a u a u a 3 uv a 4 uv, (3) v t D v = a 5 uv a 6 v a 7 v + a 8 uv. (4) The added terms a 4 uv, and a 8 uv mean that there exists an interaction between the species itself such as (in the mating period so we can consider the term v is the existence of a male and a female together, this will lead two predators

3 Existence, uniqueness solution of a modified predator-prey model 67 to meet on a prey. As the predator has a strong rapacity and can take more than it needs so it is better to consider the term uv ), and then interact the two species together in the environmental. The constants a,a,,a 8 are positive and D, D are the diffusion coefficients which are small positive constants. Finally assume that the initial data are in the same L p space for some p >, The initial values for Eqs. (3) and (4) will be given by u(x, ) = g(x), v(x, ) = h(x) respectively, where both g(x) and h(x) L p ( ). In addition we will consider small values of time t, since we are looking for local solution in the usual diffusion, but for large values of time one should talk about long range diffusion as in [, 5]. 3. Usual diffusion with p = q in the L p,q norms To easily solve Eqs. (3) and (4) we shall make the terms a u and a 6 v disappear from Eqs. (3) and (4); to do this let u(x, t) = e αt w(x, t), and v(x, t) = e βt z(x, t) where α = a, β = a 6. Therefore Eqs. (3) and (4) together with the initial data become as follows: w t D w = a e αt w a 3 e βt wz a 4 e βt wz, (5) w(x, ) = g(x), x, (6) z t D z = a 5 e αt wz a 7 e βt z + a 8 e (α+β)t wz, (7) z(x, ) = h(x), x, (8) since we have the heat operator in the left hand side of Eqs.(5) and (7), t therefore w and z can be obtained by solving the following integral equations: t w = K (x y, t τ)[ a e ατ w a 3 e βτ wz a 4 e βτ wz ] dydτ t + K (x y, t) g(y)dy, (9) z = K (x y, t τ)[a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] dydτ + K (x y, t) h(y)dy, () where K, K are the fundamental solutions of the heat equation; thus: K (x, t) = D t and K πt (x, t) = D t, x = (x πt + x ), t >. Let D = max{d, D } where D is a small positive constant. And K, K can be estimated as in []; then

4 67 M. A. Al Qudah K, (x, t) D ( x +t ). () Using the symbol to represent the convolution in space and time while the symbol is to represent the convolution in space only; we can rewrite Eqs. (9) and () in a simpler way as follows: w = K [ a e ατ w a 3 e βτ wz a 4 e βτ wz ] + K g, () z = K [a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] + K h. (3) where w and z are weak solutions of Eqs. (), (), (3) and (4) respectively, which implies that the integrals in Eqs. (9) and () exist in the Lebesgue sense. Lemma If w(x, t), z(x, t)εl ( 6, 6 ) ( [,]); and g(x), h(x)εl ( 4 ) ( ), then for ε >, T(w) 6, 6 T(z) 6, 6 C 6 + C 6, 6, 6 A 6 + A 6, 6, 6 + C 6 z 8, 8 z 6, 6 z 8, 8, 6 z 6, 6 + C 4 A 6 + A 4 g 4, and z 8, 8 h 4. Proof : Consider T(w) and T(w) to be the image of w and z respectively, such that: T(w) = K [ a e ατ w a 3 e βτ wz a 4 e βτ wz ] + K g, (4) T(z) = K [a 5 e ατ wz a 7 e βτ z + a 8 e (α+β)τ wz ] + K h. (5) Assume that t takes small values in order to show the existence and uniqueness of local solutions to Eqs. (4) and (5). Take the first, second, third, and fourth terms on the right hand side of Eq. (5) we shall use exponents r, s, p, q respectively, when considering the L p norm and we have the same argument for Eq. (4). It is obvious from Eq. () that: D K (x, t) ( x + t ; (6) ) using Eq. (6) and the same estimation of K as in [7] which is given by: K (x, t) Dε x θ t (θ+ε) (7)

5 Existence, uniqueness solution of a modified predator-prey model 673 where t. T is very small, < θ < and the very small positive constant ε is chosen such that < θ + ε <. From Eqs. (4) and (7), we obtain: T(z(x, t)) Da 5 e αt ε +Da 8 e (α+β)t ε Da 7 e αt ε w(y, τ) z(y, τ) dydτ x y θ t τ (θ+ε) w(y,τ) z(y,τ) dydτ x y θ t τ (θ+ε) z(y, τ) dydτ x y θ t τ (θ+ε) + K h. (8) Now, assume that hεl p ( ), z εl q ( ), w z εl q ( ), wz εl q ( ). Let r > be chosen such that: r = p θ ; < p < 4 θ, (9) by taking the L r ( ) norm of the both sides and use the Benedek-Panzone Potential Theorem, see [8], the first, the second and the third terms of the right hand side of Eq. (8) become: T(z(., t)) r Da 5 d p ε w(., τ) p z(., τ) p dτ t τ (θ+ε) Da 7 d p ε z(., τ) p dτ + Da 8 d p ε w(., τ) p z(., τ) p dτ t τ (θ+ε) t τ (θ+ε) + K h(., t) r. () Let w(., τ) p z(., τ) p εl q (R + ), z(., τ) p εl q (R + ), and w(., τ) p z(., τ) p εl q (R + ), and lets > such that for the third term we have: s = 3 (θ + ε) 6 ; 3 < q < q (θ + ε), () Again, by applying the Benedek-Panzone Potential Theorem, the first, the second and the third terms of the right hand side of Eq. (), and taking the L s (R + )norm of the both sides, we obtain: T(z) r,s Da 5 d p d q w r,s z p,q Da 7 d p d q z p,q + Da 8 d p d q w p,q z p,q + K h r. () Now, take p = r, then from Eq.(9) we get p = and q = s, then from Eq.() θ 4 we have q =, < θ <, ε >. In addition we shall require that p = q. (θ+ε)

6 674 M. A. Al Qudah Therefor θ = ; which implies that: = q = r = s = 6, and selecting 3 A 6 = Da 5 d p d q, A 6 = Da 7 d p d q, and A 6 = Da 8 d p d q. And since K h 6 follows directly from imbedding Lemma (namely Lemma ) for the A 4 h 4 initial data, this turns will conclude the main Theorem. Lemma Let F(x, t) = K h. Assume that h L r ( ), < r <. If for p, q > with + =, then F(x, t) p q r Lp,q ( R + ) and F p,q c(r) h r, where c(r) is a constant depending on r and the dimension. Proof see [4]. Theorem 3 Suppose that the initial data g, h L(R 4 ), < r <, < θ <, if ε > is very small such that g 4 h 4 < ε, then a unique solution u, v such that u 6, 6, v 6, 6 < ; where < ε <. The proof of this theorem can be done using the same argument as the proof of Lemma. 4. Stability Analysis In this section we are discuss the stability of the modified model given by Eqs. (3), (4) in one dimension. First we dimensionless the system by setting U = a u, V = a 3 v, T = a a a t, X = ( a ) x, D = D, k D D = a 5, k a = a 6, a a k 4 = a 3, a a 7 = a 3 k, a 8 = a 3a 5, then we have: a U T = DU XX + U U UV UV, V T = k V[U k V + UV] + V XX. (3) It is clear that k, k are positive constants since a, a,, a 8 are positive using the fact that only the predator are capable of moving toward the prey we have D = and letting U(X, T) = u (ξ) and V(X, T) = v (ξ): ξ = X ct where c is the wave speed which must be determined, thus system (3) become: c du dξ = u [ u v v ], c dv dξ = k v [u k v u v ] + d v dξ, (4) Using the method of reduction of order by setting m= dv then we have a system of dξ nonlinear of first order ODE's:

7 Existence, uniqueness solution of a modified predator-prey model 675 du dξ = u c [ u v v ], dv dξ = m, dm dξ = cm k v [u k v u v ]. (5) It is clear that system (5) is an almost linear system and has equilibrium points of the form (u, v, m) by setting du = dv = dm = the equilibrium points dξ dξ dξ are:(,,), (,,) and (, k, ). The Jacobian matrix corresponding system (5) is: [ u v v ] [u +u v ] c c J = [ ]. k v + k v k u + k k + k v k u v c It is clear that the points (,,) and (,,) are unstable saddle points since the corresponding eigenvalues of the characteristic equation for each equilibria are of opposite signs, but the point (, k, ) is asymptotically stable because its characteristic equation: ( k k c roots: λ = k k c λ) (λ + cλ + k k ) = has the following < if k ε [, + 5 ] since k and c are positive constants. λ,3 = c± c 4k k < if c > k k then the point is node, and if c < k k then the point is spiral in each case the point is asymptotically stable. Conclusion A- The wave speed c depends on the a, a, a 5 and a 6 actually on a 4 and a 8 (since a 5 = a 8 a 4 a 3 ) which are related to the coefficients of uv and uv. Thus the coefficients of uv in the prey equation is less than the coefficient of uv, which means that, the decay of the prey decreases after we added the term uv. Also in the predator equation we have added the term uv with rate a 8, and a 8 > a 5, which means that, the growth rate of the predator increases after we add the term uv. Therefore the interaction between the predator and the prey with the term uv is better than the interaction with the term uv. B- The interaction between two predators and one prey is possible even though we are at closed environment that is for several reasons: )i) The temperature has an effect on movement of the predator and prey. For example if it increases, the predator's movement will slow down. And if it decreases the prey will hide. Consequently, two predators will gather on a prey.

8 676 M. A. Al Qudah (ii) In the mating period we can consider the term v is the existence of a male and a female together, this will lead two predators to meet on a prey. As the predator has a strong rapacity and can take more than his needs so it is better to consider the term uv. In both cases, low temperature and mating, the consumption of food will decrease and providing food will be continuous to all the species. References [] M. S. Abualrub, Diffusion Problems in Mathematical Biology, Dirasat Natural and Engineering Sciences, 3 (996), no., 6-5. [] M. S. Abualrub, Long Range Diffusion-Reaction Model on Population Dynamics, Documenta Mathematica, 3 (998), [3] M. S. Abualrub, Traveling Wave Solutions to Predator-Prey and Insect Dispersal Models, Accepted for Publication in the Journal of Franklin Institute, USA. (5). [4] M. S. Abualrub, Existence and Uniqueness of Solutions to a Diffusive Predator-Prey Model, Journal of Applied Functional Analysis, 4 (9), no., 7-. [5] M. A. Al-Qudah, and M. S. Abualrub, Existence of Solutions to a Model of Long Range Diffusion Involving Flux, Journal of Applied Functional Analysis, 5 (), no. 4, [6] M. A. Al-Qudah, Stability of Solutions for Some Non-Linear Partial Differential Equations, PhD.Thesis, University of Jordan,. [7] M. A. Al-Qudah, and M. S. Abualrub, Solitary and traveling wave solutions to a model of long range diffusion involving flux with stability analysis, International Mathematical Forum, 6 (), no. 7, [8] A. Benedek and R. Panzone, The space L^pwith mixed norm, Duke Math. J., 8 (96), [9] C.P. Clarderon, Diffusion and nonlinear population Theory, Rev. U. Mat., 35 (99), [] P. C. Fife, Stationary Patterns for Reaction-Diffusion Equations, Pitman Research Notes in Mathematics, Eds, W. E. Fitzgibbon, III and H. W. Walker, 4 (977), 8-.

9 Existence, uniqueness solution of a modified predator-prey model 677 [] S.A. Levin, and L.A. Segel, Hypothesis for the Origin of Planktonic Patchiness, Nature, 59 (976), [] A. J. Lotka, Elements of Mathematical Biology, New York Dover Pub, 956. Received: September 8, 6; Published: November 5, 6

Stability Analysis of Plankton Ecosystem Model. Affected by Oxygen Deficit

Stability Analysis of Plankton Ecosystem Model. Affected by Oxygen Deficit Applied Mathematical Sciences Vol 9 2015 no 81 4043-4052 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/1012988/ams201553255 Stability Analysis of Plankton Ecosystem Model Affected by Oxygen Deficit Yuriska

More information

Global Stability Analysis on a Predator-Prey Model with Omnivores

Global Stability Analysis on a Predator-Prey Model with Omnivores Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani

More information

Dynamical Analysis of a Harvested Predator-prey. Model with Ratio-dependent Response Function. and Prey Refuge

Dynamical Analysis of a Harvested Predator-prey. Model with Ratio-dependent Response Function. and Prey Refuge Applied Mathematical Sciences, Vol. 8, 214, no. 11, 527-537 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/12988/ams.214.4275 Dynamical Analysis of a Harvested Predator-prey Model with Ratio-dependent

More information

Variational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation

Variational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation International Journal of Mathematical Analysis Vol. 11, 2017, no. 21, 1007-1018 HIKAI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.710141 Variational Theory of Solitons for a Higher Order Generalized

More information

Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients

Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable

More information

Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay

Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Applied Mathematical Sciences, Vol 11, 2017, no 22, 1089-1095 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/ams20177271 Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Luca Guerrini

More information

Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation

Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type

More information

Weak Solutions to Nonlinear Parabolic Problems with Variable Exponent

Weak Solutions to Nonlinear Parabolic Problems with Variable Exponent International Journal of Mathematical Analysis Vol. 1, 216, no. 12, 553-564 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.216.6223 Weak Solutions to Nonlinear Parabolic Problems with Variable

More information

Solving Homogeneous Systems with Sub-matrices

Solving Homogeneous Systems with Sub-matrices Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State

More information

The Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations

The Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations Nonlinear Analysis and Differential Equations, Vol. 3, 015, no. 3, 111-1 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/nade.015.416 The Modified Adomian Decomposition Method for Solving Nonlinear

More information

KKM-Type Theorems for Best Proximal Points in Normed Linear Space

KKM-Type Theorems for Best Proximal Points in Normed Linear Space International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed

More information

Secure Weakly Connected Domination in the Join of Graphs

Secure Weakly Connected Domination in the Join of Graphs International Journal of Mathematical Analysis Vol. 9, 2015, no. 14, 697-702 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.519 Secure Weakly Connected Domination in the Join of Graphs

More information

A Class of Multi-Scales Nonlinear Difference Equations

A Class of Multi-Scales Nonlinear Difference Equations Applied Mathematical Sciences, Vol. 12, 2018, no. 19, 911-919 HIKARI Ltd, www.m-hiari.com https://doi.org/10.12988/ams.2018.8799 A Class of Multi-Scales Nonlinear Difference Equations Tahia Zerizer Mathematics

More information

Travelling waves. Chapter 8. 1 Introduction

Travelling waves. Chapter 8. 1 Introduction Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part

More information

Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces

Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces International Journal of Mathematical Analysis Vol. 11, 2017, no. 6, 267-275 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.717 Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric

More information

Stability Analysis of a Continuous Model of Mutualism with Delay Dynamics

Stability Analysis of a Continuous Model of Mutualism with Delay Dynamics International Mathematical Forum, Vol. 11, 2016, no. 10, 463-473 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.616 Stability Analysis of a Continuous Model of Mutualism with Delay Dynamics

More information

Continuous Threshold Policy Harvesting in Predator-Prey Models

Continuous Threshold Policy Harvesting in Predator-Prey Models Continuous Threshold Policy Harvesting in Predator-Prey Models Jon Bohn and Kaitlin Speer Department of Mathematics, University of Wisconsin - Madison Department of Mathematics, Baylor University July

More information

Third and Fourth Order Piece-wise Defined Recursive Sequences

Third and Fourth Order Piece-wise Defined Recursive Sequences International Mathematical Forum, Vol. 11, 016, no., 61-69 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.016.5973 Third and Fourth Order Piece-wise Defined Recursive Sequences Saleem Al-Ashhab

More information

Restrained Weakly Connected Independent Domination in the Corona and Composition of Graphs

Restrained Weakly Connected Independent Domination in the Corona and Composition of Graphs Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 973-978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121046 Restrained Weakly Connected Independent Domination in the Corona and

More information

On the Solution of the n-dimensional k B Operator

On the Solution of the n-dimensional k B Operator Applied Mathematical Sciences, Vol. 9, 015, no. 10, 469-479 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.1988/ams.015.410815 On the Solution of the n-dimensional B Operator Sudprathai Bupasiri Faculty

More information

Nonlinear Autonomous Dynamical systems of two dimensions. Part A

Nonlinear Autonomous Dynamical systems of two dimensions. Part A Nonlinear Autonomous Dynamical systems of two dimensions Part A Nonlinear Autonomous Dynamical systems of two dimensions x f ( x, y), x(0) x vector field y g( xy, ), y(0) y F ( f, g) 0 0 f, g are continuous

More information

Stability Analysis and Numerical Solution for. the Fractional Order Biochemical Reaction Model

Stability Analysis and Numerical Solution for. the Fractional Order Biochemical Reaction Model Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 11, 51-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.6531 Stability Analysis and Numerical Solution for the Fractional

More information

Secure Weakly Convex Domination in Graphs

Secure Weakly Convex Domination in Graphs Applied Mathematical Sciences, Vol 9, 2015, no 3, 143-147 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams2015411992 Secure Weakly Convex Domination in Graphs Rene E Leonida Mathematics Department

More information

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:

More information

Poincaré`s Map in a Van der Pol Equation

Poincaré`s Map in a Van der Pol Equation International Journal of Mathematical Analysis Vol. 8, 014, no. 59, 939-943 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.411338 Poincaré`s Map in a Van der Pol Equation Eduardo-Luis

More information

On the Deformed Theory of Special Relativity

On the Deformed Theory of Special Relativity Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 6, 275-282 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.61140 On the Deformed Theory of Special Relativity Won Sang Chung 1

More information

Generalized Boolean and Boolean-Like Rings

Generalized Boolean and Boolean-Like Rings International Journal of Algebra, Vol. 7, 2013, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.2894 Generalized Boolean and Boolean-Like Rings Hazar Abu Khuzam Department

More information

Diophantine Equations. Elementary Methods

Diophantine Equations. Elementary Methods International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,

More information

Math 312 Lecture Notes Linearization

Math 312 Lecture Notes Linearization Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior

More information

A Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique

A Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique Int. Journal of Math. Analysis, Vol. 7, 3, no. 53, 65-636 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ijma.3.3894 A Study on Linear and Nonlinear Stiff Problems Using Single-Term Haar Wavelet Series

More information

Hopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd

Hopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd Hopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd College of William and Mary Williamsburg, Virginia 23187 Mathematical Applications in Ecology and Evolution Workshop

More information

Problem set 7 Math 207A, Fall 2011 Solutions

Problem set 7 Math 207A, Fall 2011 Solutions Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase

More information

On a Certain Representation in the Pairs of Normed Spaces

On a Certain Representation in the Pairs of Normed Spaces Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida

More information

Weighted Composition Followed by Differentiation between Weighted Bergman Space and H on the Unit Ball 1

Weighted Composition Followed by Differentiation between Weighted Bergman Space and H on the Unit Ball 1 International Journal of Mathematical Analysis Vol 9, 015, no 4, 169-176 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ijma015411348 Weighted Composition Followed by Differentiation between Weighted

More information

Solitary Wave Solution of the Plasma Equations

Solitary Wave Solution of the Plasma Equations Applied Mathematical Sciences, Vol. 11, 017, no. 39, 1933-1941 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.7609 Solitary Wave Solution of the Plasma Equations F. Fonseca Universidad Nacional

More information

Existence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions

Existence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without

More information

3.5 Competition Models: Principle of Competitive Exclusion

3.5 Competition Models: Principle of Competitive Exclusion 94 3. Models for Interacting Populations different dimensional parameter changes. For example, doubling the carrying capacity K is exactly equivalent to halving the predator response parameter D. The dimensionless

More information

Alternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations

Alternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations International Mathematical Forum, Vol. 9, 2014, no. 35, 1725-1739 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.410170 Alternate Locations of Equilibrium Points and Poles in Complex

More information

Models Involving Interactions between Predator and Prey Populations

Models Involving Interactions between Predator and Prey Populations Models Involving Interactions between Predator and Prey Populations Matthew Mitchell Georgia College and State University December 30, 2015 Abstract Predator-prey models are used to show the intricate

More information

On Uniform Limit Theorem and Completion of Probabilistic Metric Space

On Uniform Limit Theorem and Completion of Probabilistic Metric Space Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 455-461 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4120 On Uniform Limit Theorem and Completion of Probabilistic Metric Space

More information

The Representation of Energy Equation by Laplace Transform

The Representation of Energy Equation by Laplace Transform Int. Journal of Math. Analysis, Vol. 8, 24, no. 22, 93-97 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ijma.24.442 The Representation of Energy Equation by Laplace Transform Taehee Lee and Hwajoon

More information

Potential Symmetries and Differential Forms. for Wave Dissipation Equation

Potential Symmetries and Differential Forms. for Wave Dissipation Equation Int. Journal of Math. Analysis, Vol. 7, 2013, no. 42, 2061-2066 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.36163 Potential Symmetries and Differential Forms for Wave Dissipation

More information

On Regular Prime Graphs of Solvable Groups

On Regular Prime Graphs of Solvable Groups International Journal of Algebra, Vol. 10, 2016, no. 10, 491-495 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.6858 On Regular Prime Graphs of Solvable Groups Donnie Munyao Kasyoki Department

More information

Remark on a Couple Coincidence Point in Cone Normed Spaces

Remark on a Couple Coincidence Point in Cone Normed Spaces International Journal of Mathematical Analysis Vol. 8, 2014, no. 50, 2461-2468 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49293 Remark on a Couple Coincidence Point in Cone Normed

More information

Contra θ-c-continuous Functions

Contra θ-c-continuous Functions International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 1, 43-50 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.714 Contra θ-c-continuous Functions C. W. Baker

More information

New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems

New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems Applied Mathematical Sciences, Vol., 207, no. 49, 2447-2457 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.207.7928 New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point

More information

Locating Chromatic Number of Banana Tree

Locating Chromatic Number of Banana Tree International Mathematical Forum, Vol. 12, 2017, no. 1, 39-45 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610138 Locating Chromatic Number of Banana Tree Asmiati Department of Mathematics

More information

Hyperbolic Functions and. the Heat Balance Integral Method

Hyperbolic Functions and. the Heat Balance Integral Method Nonl. Analysis and Differential Equations, Vol. 1, 2013, no. 1, 23-27 HIKARI Ltd, www.m-hikari.com Hyperbolic Functions and the Heat Balance Integral Method G. Nhawu and G. Tapedzesa Department of Mathematics,

More information

A Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings

A Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive

More information

Math 128A Spring 2003 Week 12 Solutions

Math 128A Spring 2003 Week 12 Solutions Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems

More information

Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists

Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists International Journal of Mathematical Analysis Vol 9, 2015, no 38, 1869-1875 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma201554135 Direction and Stability of Hopf Bifurcation in a Delayed Model

More information

Functional Response to Predators Holling type II, as a Function Refuge for Preys in Lotka-Volterra Model

Functional Response to Predators Holling type II, as a Function Refuge for Preys in Lotka-Volterra Model Applied Mathematical Sciences, Vol. 9, 2015, no. 136, 6773-6781 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.53266 Functional Response to Predators Holling type II, as a Function Refuge

More information

Continuous time population models

Continuous time population models Continuous time population models Jaap van der Meer jaap.van.der.meer@nioz.nl Abstract Many simple theoretical population models in continuous time relate the rate of change of the size of two populations

More information

On Two New Classes of Fibonacci and Lucas Reciprocal Sums with Subscripts in Arithmetic Progression

On Two New Classes of Fibonacci and Lucas Reciprocal Sums with Subscripts in Arithmetic Progression Applied Mathematical Sciences Vol. 207 no. 25 2-29 HIKARI Ltd www.m-hikari.com https://doi.org/0.2988/ams.207.7392 On Two New Classes of Fibonacci Lucas Reciprocal Sums with Subscripts in Arithmetic Progression

More information

A Generalization of p-rings

A Generalization of p-rings International Journal of Algebra, Vol. 9, 2015, no. 8, 395-401 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5848 A Generalization of p-rings Adil Yaqub Department of Mathematics University

More information

Approximation to the Dissipative Klein-Gordon Equation

Approximation to the Dissipative Klein-Gordon Equation International Journal of Mathematical Analysis Vol. 9, 215, no. 22, 159-163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.5236 Approximation to the Dissipative Klein-Gordon Equation Edilber

More information

Instabilities In A Reaction Diffusion Model: Spatially Homogeneous And Distributed Systems

Instabilities In A Reaction Diffusion Model: Spatially Homogeneous And Distributed Systems Applied Mathematics E-Notes, 10(010), 136-146 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Instabilities In A Reaction Diffusion Model: Spatially Homogeneous And

More information

Qualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions

Qualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 1, 45-59 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.3819 Qualitative Theory of Differential Equations and Dynamics of

More information

Antibound State for Klein-Gordon Equation

Antibound State for Klein-Gordon Equation International Journal of Mathematical Analysis Vol. 8, 2014, no. 59, 2945-2949 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.411374 Antibound State for Klein-Gordon Equation Ana-Magnolia

More information

Weyl s Theorem and Property (Saw)

Weyl s Theorem and Property (Saw) International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, 433-437 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.8754 Weyl s Theorem and Property (Saw) N. Jayanthi Government

More information

Numerical Solution of Heat Equation by Spectral Method

Numerical Solution of Heat Equation by Spectral Method Applied Mathematical Sciences, Vol 8, 2014, no 8, 397-404 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201439502 Numerical Solution of Heat Equation by Spectral Method Narayan Thapa Department

More information

On Generalized Derivations and Commutativity. of Prime Rings with Involution

On Generalized Derivations and Commutativity. of Prime Rings with Involution International Journal of Algebra, Vol. 11, 2017, no. 6, 291-300 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7839 On Generalized Derivations and Commutativity of Prime Rings with Involution

More information

Remarks on the Maximum Principle for Parabolic-Type PDEs

Remarks on the Maximum Principle for Parabolic-Type PDEs International Mathematical Forum, Vol. 11, 2016, no. 24, 1185-1190 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2016.69125 Remarks on the Maximum Principle for Parabolic-Type PDEs Humberto

More information

Lotka Volterra Predator-Prey Model with a Predating Scavenger

Lotka Volterra Predator-Prey Model with a Predating Scavenger Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics

More information

A Solution of the Spherical Poisson-Boltzmann Equation

A Solution of the Spherical Poisson-Boltzmann Equation International Journal of Mathematical Analysis Vol. 1, 018, no. 1, 1-7 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ijma.018.71155 A Solution of the Spherical Poisson-Boltzmann quation. onseca

More information

An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh

An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh International Mathematical Forum, Vol. 8, 2013, no. 9, 427-432 HIKARI Ltd, www.m-hikari.com An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh Richard F. Ryan

More information

k-weyl Fractional Derivative, Integral and Integral Transform

k-weyl Fractional Derivative, Integral and Integral Transform Int. J. Contemp. Math. Sciences, Vol. 8, 213, no. 6, 263-27 HIKARI Ltd, www.m-hiari.com -Weyl Fractional Derivative, Integral and Integral Transform Luis Guillermo Romero 1 and Luciano Leonardo Luque Faculty

More information

A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage

A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and

More information

Restrained Independent 2-Domination in the Join and Corona of Graphs

Restrained Independent 2-Domination in the Join and Corona of Graphs Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs

More information

Optimal Homotopy Asymptotic Method for Solving Gardner Equation

Optimal Homotopy Asymptotic Method for Solving Gardner Equation Applied Mathematical Sciences, Vol. 9, 2015, no. 53, 2635-2644 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52145 Optimal Homotopy Asymptotic Method for Solving Gardner Equation Jaharuddin

More information

A Recursion Scheme for the Fisher Equation

A Recursion Scheme for the Fisher Equation Applied Mathematical Sciences, Vol. 8, 204, no. 08, 536-5368 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.204.47570 A Recursion Scheme for the Fisher Equation P. Sitompul, H. Gunawan,Y. Soeharyadi

More information

Why Bellman-Zadeh Approach to Fuzzy Optimization

Why Bellman-Zadeh Approach to Fuzzy Optimization Applied Mathematical Sciences, Vol. 12, 2018, no. 11, 517-522 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8456 Why Bellman-Zadeh Approach to Fuzzy Optimization Olga Kosheleva 1 and Vladik

More information

Hyers-Ulam-Rassias Stability of a Quadratic-Additive Type Functional Equation on a Restricted Domain

Hyers-Ulam-Rassias Stability of a Quadratic-Additive Type Functional Equation on a Restricted Domain Int. Journal of Math. Analysis, Vol. 7, 013, no. 55, 745-75 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.013.394 Hyers-Ulam-Rassias Stability of a Quadratic-Additive Type Functional Equation

More information

Direct Product of BF-Algebras

Direct Product of BF-Algebras International Journal of Algebra, Vol. 10, 2016, no. 3, 125-132 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.614 Direct Product of BF-Algebras Randy C. Teves and Joemar C. Endam Department

More information

International Mathematical Forum, Vol. 9, 2014, no. 36, HIKARI Ltd,

International Mathematical Forum, Vol. 9, 2014, no. 36, HIKARI Ltd, International Mathematical Forum, Vol. 9, 2014, no. 36, 1751-1756 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.411187 Generalized Filters S. Palaniammal Department of Mathematics Thiruvalluvar

More information

Math 5490 November 5, 2014

Math 5490 November 5, 2014 Math 549 November 5, 214 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2:3 3:45 http://www.math.umn.edu/~mcgehee/teaching/math549-214-2fall/ Streaming

More information

Dynamical Behavior for Optimal Cubic-Order Multiple Solver

Dynamical Behavior for Optimal Cubic-Order Multiple Solver Applied Mathematical Sciences, Vol., 7, no., 5 - HIKARI Ltd, www.m-hikari.com https://doi.org/.988/ams.7.6946 Dynamical Behavior for Optimal Cubic-Order Multiple Solver Young Hee Geum Department of Applied

More information

Prime and Semiprime Bi-ideals in Ordered Semigroups

Prime and Semiprime Bi-ideals in Ordered Semigroups International Journal of Algebra, Vol. 7, 2013, no. 17, 839-845 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.310105 Prime and Semiprime Bi-ideals in Ordered Semigroups R. Saritha Department

More information

Solution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods

Solution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the

More information

A Note on the Variational Formulation of PDEs and Solution by Finite Elements

A Note on the Variational Formulation of PDEs and Solution by Finite Elements Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 173-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8412 A Note on the Variational Formulation of PDEs and Solution by

More information

An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson Method

An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson Method Applied Mathematical Sciences, Vol. 11, 2017, no. 56, 2789-2797 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.710302 An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations Michael H. F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands December 2005 Overview What are Ordinary Differential Equations

More information

Novel Approach to Calculation of Box Dimension of Fractal Functions

Novel Approach to Calculation of Box Dimension of Fractal Functions Applied Mathematical Sciences, vol. 8, 2014, no. 144, 7175-7181 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49718 Novel Approach to Calculation of Box Dimension of Fractal Functions

More information

1.Introduction: 2. The Model. Key words: Prey, Predator, Seasonality, Stability, Bifurcations, Chaos.

1.Introduction: 2. The Model. Key words: Prey, Predator, Seasonality, Stability, Bifurcations, Chaos. Dynamical behavior of a prey predator model with seasonally varying parameters Sunita Gakkhar, BrhamPal Singh, R K Naji Department of Mathematics I I T Roorkee,47667 INDIA Abstract : A dynamic model based

More information

Improvements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method

Improvements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear

More information

9 More on the 1D Heat Equation

9 More on the 1D Heat Equation 9 More on the D Heat Equation 9. Heat equation on the line with sources: Duhamel s principle Theorem: Consider the Cauchy problem = D 2 u + F (x, t), on x t x 2 u(x, ) = f(x) for x < () where f

More information

On Strong Alt-Induced Codes

On Strong Alt-Induced Codes Applied Mathematical Sciences, Vol. 12, 2018, no. 7, 327-336 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8113 On Strong Alt-Induced Codes Ngo Thi Hien Hanoi University of Science and

More information

A Stochastic Viral Infection Model with General Functional Response

A Stochastic Viral Infection Model with General Functional Response Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 9, 435-445 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.664 A Stochastic Viral Infection Model with General Functional Response

More information

Finite Difference Method of Fractional Parabolic Partial Differential Equations with Variable Coefficients

Finite Difference Method of Fractional Parabolic Partial Differential Equations with Variable Coefficients International Journal of Contemporary Mathematical Sciences Vol. 9, 014, no. 16, 767-776 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.1988/ijcms.014.411118 Finite Difference Method of Fractional Parabolic

More information

Symmetry Reduction of Chazy Equation

Symmetry Reduction of Chazy Equation Applied Mathematical Sciences, Vol 8, 2014, no 70, 3449-3459 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201443208 Symmetry Reduction of Chazy Equation Figen AÇIL KİRAZ Department of Mathematics,

More information

Double Total Domination in Circulant Graphs 1

Double Total Domination in Circulant Graphs 1 Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye

More information

Figure 1: Ca2+ wave in a Xenopus oocyte following fertilization. Time goes from top left to bottom right. From Fall et al., 2002.

Figure 1: Ca2+ wave in a Xenopus oocyte following fertilization. Time goes from top left to bottom right. From Fall et al., 2002. 1 Traveling Fronts Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 When mature Xenopus oocytes (frog

More information

Quadratic Optimization over a Polyhedral Set

Quadratic Optimization over a Polyhedral Set International Mathematical Forum, Vol. 9, 2014, no. 13, 621-629 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.4234 Quadratic Optimization over a Polyhedral Set T. Bayartugs, Ch. Battuvshin

More information

An analogue of Rionero s functional for reaction-diffusion equations and an application thereof

An analogue of Rionero s functional for reaction-diffusion equations and an application thereof Note di Matematica 7, n., 007, 95 105. An analogue of Rionero s functional for reaction-diffusion equations and an application thereof James N. Flavin Department of Mathematical Physics, National University

More information

Research on Independence of. Random Variables

Research on Independence of. Random Variables Applied Mathematical Sciences, Vol., 08, no. 3, - 7 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/ams.08.8708 Research on Independence of Random Variables Jian Wang and Qiuli Dong School of Mathematics

More information

Weak Resolvable Spaces and. Decomposition of Continuity

Weak Resolvable Spaces and. Decomposition of Continuity Pure Mathematical Sciences, Vol. 6, 2017, no. 1, 19-28 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/pms.2017.61020 Weak Resolvable Spaces and Decomposition of Continuity Mustafa H. Hadi University

More information

THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL

THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL HAL L. SMITH* SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES ARIZONA STATE UNIVERSITY TEMPE, AZ, USA 8587 Abstract. This is intended as lecture notes for nd

More information

A Two-step Iterative Method Free from Derivative for Solving Nonlinear Equations

A Two-step Iterative Method Free from Derivative for Solving Nonlinear Equations Applied Mathematical Sciences, Vol. 8, 2014, no. 161, 8021-8027 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49710 A Two-step Iterative Method Free from Derivative for Solving Nonlinear

More information

Generalization of the Banach Fixed Point Theorem for Mappings in (R, ϕ)-spaces

Generalization of the Banach Fixed Point Theorem for Mappings in (R, ϕ)-spaces International Mathematical Forum, Vol. 10, 2015, no. 12, 579-585 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.5861 Generalization of the Banach Fixed Point Theorem for Mappings in (R,

More information

Introduction LECTURE 1

Introduction LECTURE 1 LECTURE 1 Introduction The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in

More information