animprovedcollocationapproachofeulerpolynomials...

Research ArticleAn Improved Collocation Approach of Euler PolynomialsConnected with Bernoulli Ones for Solving Predator-PreyModels with Time Lag

Behrooz Basirat and Hamid Reza Elahi

Department of Mathematics Birjand Branch Islamic Azad University Birjand Iran

Correspondence should be addressed to Behrooz Basirat behroozbasiratiaubiracir

Received 1 September 2019 Revised 6 November 2019 Accepted 12 December 2019 Published 1 April 2020

Academic Editor Patricia J Y Wong

Copyright copy 2020 Behrooz Basirat and Hamid Reza Elahi is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

is paper deals with an approach to obtaining the numerical solution of the LotkandashVolterra predator-prey models with discretedelay using Euler polynomials connected with Bernoulli ones By using the Euler polynomials connected with Bernoulli ones andcollocation points this method transforms the predator-prey model into a matrix equation e main characteristic of thisapproach is that it reduces the predator-prey model to a system of algebraic equations which greatly simplifies the problem Forthese models the explicit formula determining the stability and the direction is given Numerical examples illustrate the reliabilityand efficiency of the proposed scheme

1 Introduction

In recent years population models in various fields ofmathematical biology have been proposed and studied ex-tensively Predator-prey interaction is the fundamentalstructure in population dynamics Understanding the dy-namics of predator-prey models will be very helpful for

investigating multiple species interactions [1ndash6] e firstpredator-prey model with delay was proposed by Volterraand Brelot [7] Let p1(t) and p2(t) denote the populationdensity of prey and predator at time t In this work we firstconsider the modified version of the predator-prey modelwith delay [8]

p1prime(t) p1(t) r1 minus a11p1(t) minus a12 1113946t

minusinfinF(t minus s)p2(s)ds1113890 1113891

p2prime(t) p2(t) minusr2 + a21 1113946t

infinG(t minus s)p1(s)ds minus a22p2(t)1113890 1113891

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)

where r1 gt 0 is the growth rate of the prey in the absence ofpredators a11 gt 0 denotes the self-regulation constant of theprey a12 gt 0 describes the predation of the prey by predatorsr2 gt 0 is the death rate of predators in the absence of the preya21 gt 0 is the conversion rate for the predators and a22 ge 0describes the transpacific competition among predatorsF(s) and G(s) are nonnegative continuous delay kernels

defined and integrable on [0infin) which weigh the contri-bution of the predation occurred in the past to the changerate of the prey and predators respectively Detailed studyon stability and bifurcation of system (1) can be found inCushing [9] When F(t minus s) δ(t minus s minus τ1) andG(t minus s)

δ(t minus s minus τ2)mdashwhere δ is the delta function and τ1 ge 0 andτ2 ge 0 are the hunting delay and the maturation time of the

HindawiInternational Journal of Differential EquationsVolume 2020 Article ID 9176784 8 pageshttpsdoiorg10115520209176784

prey species respectivelymdashsystem (1) reduces to the fol-lowing LotkandashVolterra predator-prey model with two dis-crete delays

p1prime(t) p1(t) r1 minus a11p1(t) minus a12p2 t minus τ1( 11138571113858 1113859

p2prime(t) p2(t) minusr2 + a21p1 t minus τ2( 1113857 minus a22p2(t)1113858 1113859

⎧⎨

⎩ (2)

We will find the approximate solutions of the system (2)with initial conditions

p1(0) α1

p2(0) α2(3)

where α1 and α2 are positive numberse exact solution for the predator-prey models in the

general form does not exist therefore numerical methodsare needed to computing the population densities of the preyand predator Somemethods [10ndash12] are proposed to handlean approximate solution of the predatorndashprey system suchas finite-difference finite element and spectral methods

e organization of this paper is as follows In Section 2we study the stability of the predator-prey model In Section3 matrix relations for Euler polynomials connected withBernoulli ones will be discussed In Section 4 we shall

present our method in detail Finally we give a few nu-merical examples in Section 5

2 Stability

In this section we review some results on the stability of thepredator-prey model To begin with we first consider system(2) appearing in the interspecific interaction terms of bothequations

p1prime(t) p1(t)f p1(t) p2 t minus τ1( 1113857( 1113857

p2prime(t) p2(t)g p1 t minus τ2( 1113857 p2(t)( 1113857

⎧⎨

⎩ (4)

where τi ge 0 i 1 2 is a constant Denote C C([minusτ

0] times R) where τ max τ1 τ21113864 1113865 Assume that f R times C⟶ R

and g R times C⟶ R satisfy the following assumptions

(S1) there exists a point (plowast1 plowast2 ) with plowast1 plowast2 gt 0 forwhich f(plowast1 plowast2 ) g(plowast1 plowast2 ) 0(S2) f and g are continuously differentiable such thatzfzp1 lt 0 zfzp2 lt 0 zgzp1 gt 0 and zgzp2 gt 0

Assumption (S1) ensures that (plowast1 plowast2 ) is a positiveequilibrium point of system (4) Let ϕ(t) p1(t) minus plowast1 andψ(t) p2(t) minus plowast2 en the linearized system at (plowast1 plowast2 ) is

ϕprime(t) zf

zp1plowast1 plowast2( 11138571113888 1113889 plowast1( 1113857ϕ(t) +

zf

zp2plowast1 plowast2( 11138571113888 1113889 plowast1( 1113857ψ t minus τ1( 1113857

ψprime(t) zg

zp1plowast1 plowast2( 11138571113888 1113889 plowast2( 1113857ϕ t minus τ2( 1113857 +

zg

zp2plowast1 plowast2( 11138571113888 1113889 plowast2( 1113857ψ(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(5)

erefore we have the following theorem

Theorem 1 Suppose that f and g satisfy the assumptions(S1) and (S2) Define

a zf

zp1plowast1 plowast2( 1113857

b zf


c zg


d zg


(6)

en the positive equilibrium (plowast1 plowast2 ) of the delayedpredator-prey system (4) is absolutely stable if and only ifa d + bcgt 0

For proof see [13]By eorem 1 we have the following condition If

r1a21 minus r2a11 gt 0 (7)

then system (2) has a positive equilibrium Elowast (plowast1 plowast2 )where

plowast1

r1a22 + r2a12

a11a22 + a12a21

plowast2

r1a21 minus r2a11

a11a22 + a12a21

(8)

e condition a d + bcgt 0 becomes a11a22 minus a12a21 gt 0us by eorem 1 we have the following result on thestability of Elowast (plowast1 plowast2 )

Corollary 1 If condition (7) is satisfied ie if the positiveequilibrium Elowast (plowast1 plowast2 ) of system (2) exists then it isabsolutely stable if and only if a11a22 minus a12a21 gt 0

2 International Journal of Differential Equations

He [14] and Lu and Wang [15] showed that the positiveequilibrium Elowast (plowast1 plowast2 ) of system (2) is globally stablee above stability result depends on the assumption thata11a22 minus a12a21 gt 0 If a11a22 minus a12a21 lt 0 then Faria [16]proved the following result

Corollary 2 If a11a22 minus a12a21 lt 0 and a11a22 ne 0 then thereis a critical value τ20 gt 0 such that Elowast of system (2) is as-ymptotically stable when τ2 lt τ20 and unstable when τ2 gt τ20

3 Matrix Relations for Euler Polynomials

Bernoulli polynomials Bn(x) are usually defined from thegenerating function for |t|lt 2π

G(x t) ≔text

et minus 1 1113944infin

n0Bn(x)

tn

n (9)

and Bernoulli numbers βn Bn(0) are contained in manymathematical formulas such as the Taylor expansion in aneighborhood of the origin of trigonometric and hyperbolictangent and cotangent functions and they can be obtainedby the generating function [17]

t


n0βn

tn

n (10)

Euler studied Bernoulli polynomials and these poly-nomials are employed in the integral representation ofdifferentiable periodic functions and play an important rolein the approximation of such functions using polynomialsMany early Euler and Bernoulli polynomial implementa-tions can be contained in [18ndash20] Euler polynomials arestrictly related to Bernoulli and are used in Taylorrsquos ex-pansion in the trigonometric and hyperbolic secant functiondistricts Recursive computation of Bernoulli and Eulerpolynomials can be obtained by using the followingformulas

1113944nminus1

k0

n

k1113888 1113889Bk(t) nt

nminus 1 n 2 3

En(t) + 1113944n

k0

n

k1113888 1113889Ek(t) 2t

n n 1 2

(11)

Euler polynomials En(t) can also be defined as poly-nomials of degree nge 0 satisfying the conditions

(1) Emprime (t) mEmminus1(t) mge 1

(2) Em(t + 1) + Em(t) 2tm mge 1

e above conditions express a recurrence relation toEuler polynomials An explicit formula for En(t) is repre-sented as

En(t) 1

n + 11113944

n+1

k12 minus 2k+1

1113872 1113873

n + 1

k

⎛⎝ ⎞⎠βktn+1minus k

(12)

where βk Bk(0) is the first Bernoulli numbers for each k

0 1 n [21]

31 Function Approximation Let H L2[0 1] and assumethat E0(t) E1(t) EN(t)1113864 1113865 be the set of Euler polyno-mials and PH span E0(t) E1(t) EN(t)1113864 1113865 and h(t) bean arbitrary element in H Since PH is a finite dimensionalvector space h(t) has the unique best approximation out ofPH such as 1113957p(t) that is

h(t) minus 1113957p(t)le h(t) minus p(t) forallp(t) isin PH (13)

Since 1113957p(t) isin PH there exist unique coefficientsp0 p1 pN such that

h(t)≃ 1113957p(t) 1113944N

n0pnEn(t) PE(t)

hprime(t)≃ 1113957pprime(t) 1113944N

n0pnEnprime(t) PEprime(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

where

P p0 p1 pN1113858 1113859

E(t) E0(t) E1(t) EN(t)1113858 1113859T

Eprime(t) DE(t)

(15)

such that

D

0 0 middot middot middot 0 0



⋮ ⋮ ⋱ ⋮ ⋮

0 0 middot middot middot N 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Note that from property (12) of Euler polynomials wecan represent E(t) in a matrix form as follows

E(t) MT(t) (17)

where

International Journal of Differential Equations 3

M

1 0 0 middot middot middot 0

2 minus 23( 11138572

2⎛⎝ ⎞⎠

2β2 1 0 middot middot middot 0

2 minus 24( 11138573

3⎛⎝ ⎞⎠

3β3

2 minus 23( 11138573

2⎛⎝ ⎞⎠

3β2 1 middot middot middot 0

⋮ ⋮ ⋮ ⋱ ⋮

2 minus 2N+2( 1113857N + 1

N + 1⎛⎝ ⎞⎠

N + 1βN+1

2 minus 2N+1( 1113857N + 1

N

⎛⎝ ⎞⎠

N + 1βN

2 minus 2N( 1113857N + 1

N minus 1⎛⎝ ⎞⎠

N + 1βNminus1 middot middot middot 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

is an (N + 1) times (N + 1) matrix and T(t)

[1 t t2 tN]T

4 Numerical Solution of the LotkandashVoltraPredator-Prey Models with Discreet DelayUsing Euler Operational Matrices

Consider the LotkandashVoltra predator-prey models withdiscreet delay (2) For implementing the Euler operationalmatrix on the predator-prey model we first approximatepi(t) and pi

prime(t) for i 1 2 by equations (14)ndash(17) asfollows

pi(t)≃PiE(t) PiMT(t)

piprime(t)≃PiEprime(t) PiDE(t) PiDMT(t)

1113896 (19)

where Pi is the unknown (N + 1) vector of coefficients

Remark 1 Recall that the Hadamard product of two ma-trices say A and B of the same dimension m times n is denotedby A ∘B defined to be an m times n-matrix in which the (i j)th

entry is equal to the product of (i j)th entries of A and B ie(A ∘B)ij AijBij

Bearing in mind the previous remark we can simplifythe delay part of system (2) to the matrix form as

E(t minus τ) MT(t minus τ) MBτT(t) (20)

where B0 I and for any τ ne 0 Bτ is the Hadamard productC ∘ G (ie the product of entry by entry ) such that for

1le jle ileN + 1 Cij i minus 1j minus 11113888 1113889 and Gij (minusτ)iminus j and for

ilt j Cij Gij 0 In other words we may evaluate entriesof Bτ as follows

Bτ(i j)

i minus 1

j minus 11113888 1113889(minusτ)iminus j ige j

0 ilt j

⎧⎪⎪⎨

⎪⎪⎩(21)

Remark 2 Note that If A [aij]mtimesn and Bptimesq are twomatrices then the Kronecker product AotimesB is defined to bethe following mp times nq block matrix

AotimesB

a11B middot middot middot a1nB

⋮ ⋱ ⋮

am1B middot middot middot amnB

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)

By the mixed-product property of a Kronecker productwe know that for matrices A B C and D which are of thesize that matrix products AC and B D are well-defined thefollowing relation holds true

(AC)otimes (BD) (AotimesB)(CotimesD) (23)

Now we have to estimate all expressions consisting theproducts of p1(t minus τ2) and p2(t minus τ1) existing in system (2)(note that for τk 0 the products pi(t minus τk)pj(t)pi(t)pj(t minus τk) and pi(t)pj(t) can be approximated inmatrix form where i j isin 1 2 ) In order to do this leti j1113864 1113865sube 1 2 iprime 3 minus i and jprime 3 minus j en by usingequations (19) and (20) we have

pi t minus τiprime( 1113857pj t minus τjprime1113872 1113873≃ PiE t minus τiprime( 1113857( 1113857 PjE t minus τjprime1113872 11138731113872 1113873

PiMBτiprimeT(t)1113874 1113875otimes PjMBτ

jprimeT(t)1113874 1113875

Pi otimesPj1113872 1113873(MotimesM) BτiprimeotimesBτ

jprime1113874 1113875(T(t)otimesT(t))

PijMB τiprime τjprime1113872 1113873T(t)

(24)

where

Pij Pi otimesPj

M MotimesM

B τiprime τjprime1113872 1113873 BτiprimeotimesBτ

jprime

T(t) T(t)otimesT(t)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)


With the above notations we can rewrite system (2) as

P1DMT(t) r1P1MT(t) minus a11P11MT(t) minus a12P12MB 0 τ1( 1113857T(t)

P2DMT(t) minusr2P2MT(t) + a21P21MB 0 τ2( 1113857T(t) minus a22P22MT(t)

⎧⎨

⎩ (26)

Suppose that

M1(t) DMT(t) minus r1MT(t)

M2(t) DMT(t) + r2MT(t)

A1(t) a12MB 0 τ1( 1113857T(t)

A2(t) minusa21MB 0 τ2( 1113857T(t)

C1(t) a11MT(t)

C2(t) a22MT(t)

(27)

en the fundamental matrix equations of system (2)can be written in the following form

PM(t) + PA(t) + 1113954PC(t) f(t) (28)

where

P P1 P21113858 1113859

M(t) M1(t) 0

0 M2(t)1113890 1113891

P P12P211113960 1113961

A(t) A1(t) 0

0 A2(t)1113890 1113891

1113954P P11P221113960 1113961

C(t) C1(t) 0

0 C2(t)1113890 1113891

f(t) [0 0]1times2

(29)

41 Implementing the Collocation Method Equation (28)gives us 2 nonlinear equations Since the number of un-knowns for each vector P1 and P2 in (15) is (N + 1) and oursystem has 4 equations the total number of unknowns is2(N + 1) then we collocate equation (28) atNNewtonndashCotes points given as

ts b

Ns s 1 2 N (30)

and hence we have equation (28) and initial conditions

PM ts( 1113857 + PA ts( 1113857 + 1113954PC ts( 1113857 f ts( 1113857 s 1 2 N

P1MT(0) α1

P2MT(0) α2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

After solving linear system (31) we get P1 and P2 Forcollocating equation (31) we have used the NewtonndashCotespoints because of their simplicity and their good utility inour implementation regarding the speed and accuracy ofanswers However we could use other points such as theGauss points ClenshawndashCurtis points and Lobatto points

5 Numerical Examples

To test the method in terms of its precision and efficacy weturn our attention to show three numerical performanceresults Analytic approximate solutions of high-order ac-curacy are presented in most of the cases roughout thecalculations the absolute error is defined by

Error(t) pi(t) minus y1(t)1113868111386811138681113868

1113868111386811138681113868 (32)

and maximum absolute error is defined by

Error Max pi ti( 1113857 minus y1 ti( 11138571113868111386811138681113868

1113868111386811138681113868 ti i

1000 i 1 2 10001113882 1113883

(33)

e computations associated with these examples wereperformed using MATLAB

Since the truncated series (17) is approximate solutionsof system (31) when the function p1N(t) p2N(t) and itsderivatives are substituted in system (2) the resultingequation must be satisfied approximately that is forts s 1 2 N

R1N ts( 1113857 p1Nprime ts( 1113857 minus p1N ts( 1113857 1 minus p2N ts( 11138571113960 1113961

11138681113868111386811138681113868

11138681113868111386811138681113868 0

R2N ts( 1113857 p2Nprime ts( 1113857 minus p2N ts( 1113857 minus1 + p1N ts( 11138571113960 1113961

11138681113868111386811138681113868

11138681113868111386811138681113868 0

⎧⎪⎨

⎪⎩

(34)

and Max |R1N(ts) minus R2N(ts)|le 10minus rs1113966 (rs positive integer)If max10minus rs 10minus r (r positive integer) is prescribed then thetruncation limitN is increased until the difference RiN(ts) ateach of the points becomes smaller than the prescribed 10minus rAll computations were carried out using MATLAB

Example 1 We consider the following LotkandashVolterrapredator-prey model with delays

p1prime(t) p1(t) 1 minus p1(t) minus p2(t minus 17)1113858 1113859

p2prime(t) p2(t) minus1 + 2p1(t minus 18) minus p2(t)1113858 1113859

⎧⎨

⎩ (35)

which has also a positive equilibrium Elowast (plowast1

(23) plowast2 (13))[22] e numerical simulations for Ex-ample 1 are shown in Figures 1 and 2


Table 1 Errors for Example 2

tR1N(t) R2N(t)

Present method Method in [23] Present method Method in [23]N 5 N 12 N 5 N 5 N 12 N 5

00 588378e minus 8 356178e minus 15 ≃[10minus 5 10minus 10] 3415153e minus 8 8881278e minus 15 ≃[10minus 5 10minus 10]











0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)

Figure 1 e stable behavior of the prey and predator populations for Example 1 to the equilibrium population points (a) e graph onp1 minus t plane of Example 1 (b) e graph on p2 minus t plane of Example 1

05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05

Figure 2 e phase graph of Example 1 in t 500 with the equilibrium points plowast1 (23) and plowast2 (13)


Example 2 Consider the LotkandashVolterra predator-preymodels with discrete delay given in [23] by

p1prime(t) p1(t) 1 minus p2(t)1113858 1113859

p2prime(t) p2(t) minus1 + p1(t)1113858 1113859

⎧⎨

⎩ (36)

with initial conditions

p1(0) 13

p2(0) 06(37)

We selected our second example from [23] whichsolved this predator-prey interaction by the Bessel col-location approach We compared our results with theBessel collocation results and the outcomes are tabulatedin Table 1 For this example too the improved Collo-cation approach of Euler polynomials connected withBernoulli ones is incisive is is because for the samenumber of basis functions it obtains better resultsTable 1 shows the present method results for Example 2 incomparison with method of [23] e superiority of Eulerpolynomials operational matrices method compared withBessel collocation approach is clear here which is becausewith the same number of basis functions we get verybetter results

Example 3 Suppose in a closed ecosystem there are only 2types of animals the predator and the prey ey form asimple food-chain where the predator species hunts the preyspecies while the prey grazes vegetation e size of the 2populations can be described by a simple system of 2nonlinear first-order differential equations

p1prime(t) p1(t) A minus Bp2(t)1113858 1113859

p2prime(t) p2(t) minusC + Dp1(t)1113858 1113859

⎧⎨

⎩ (38)

which a set of fixed positive constants A the growth rate ofthe prey B the rate at which predators destroy the prey Cthe death rate of predators and D the rate at whichpredators increase by consuming the prey

A prey population p1(t) increases at a rate (dp1(t)dt)

Ap1(t) (proportional to the number of prey) but is si-multaneously destroyed by predators at a rate (dp1(t)dt)

minusBp1(t)p2(t) (proportional to the product of the number ofprey and predators) A predator population p2(t) decreasesat a rate (dp2(t)dt) minusCp2(t) (proportional to the numberof predators) but increases at a rate (dp2(t)dt)

Dp1(t)p2(t) (again proportional to the product of thenumbers of prey and predators) By eorem 1 we have thethe positive equilibrium Elowast (plowast1 (CD) plowast2 (AB))

For this example the stability of each of the three steadystates can be assessed more formally using the approachdiscussed e absolute errors for N 7 9 are estimated byR1N and R2N and are presented in Tables 2 and 3 We seethat if N increases then the absolute errors decrease morerapidly

6 Conclusion

In this paper we have proposed a numerical approach forsolving the LotkandashVolterra predator-prey models withdiscrete delay by utilizing Euler polynomials connected withBernoulli ones e cognate matrices and mixed-productproperty of Kronecker products besides the collocationmethod have been utilized for transforming a predator-preysystem to a linear system of algebraic equations that can besolved facilely To the best of our cognizance this is the firstwork coalescing the Euler polynomial connected with theBernoulli polynomial and collocation points for solvingLotkandashVolterra predator-prey models Finally numericalexamples reveal that the present method is very precise andconvenient for solving predator-prey models

Table 2 Comparison of the absolute errors obtained by different rates for R1N in Example 3

tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015

N 7 N 9 N 7 N 9 N 7 N 902 375745e minus 12 130547e minus 14 234442e minus 11 212354e minus 13 165098e minus 11 277903e minus 1504 100987e minus 12 322244e minus 14 102348e minus 11 677926e minus 14 130007e minus 11 599001e minus 1306 643322e minus 13 991158e minus 15 482091e minus 13 466258e minus 15 820989e minus 13 556193e minus 1508 449987e minus 11 900302e minus 14 335303e minus 11 639937e minus 15 896511e minus 11 409824e minus 1510 266561e minus 10 921367e minus 14 612546e minus 11 909012e minus 15 536768e minus 11 209891e minus 15


tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] N H Aljahdaly and M A Alqudah ldquoAnalytical solutions of amodified predator-prey model through a new ecological in-teractionrdquo Computational and Mathematical Methods inMedicine vol 2019 Article ID 4849393 7 pages 2019

[2] A Das and M Pal ldquoeoretical analysis of an imprecise prey-predator model with harvesting and optimal controlrdquo Journalof Optimization vol 2019 Article ID 9512879 12 pages 2019

[3] Q Han W Xu B Hu D Huang and J-Q Sun ldquoExtinctiontime of a stochastic predator-prey model by the generalizedcell mappingmethodrdquo Physica A Statistical Mechanics and ItsApplications vol 494 pp 351ndash366 2018

[4] C Leal-Ramırez O Castillo P Melin and H Echavarria-Heras ldquoA fuzzy cellular preyndashpredator model for pest controlunder sustainable bio-economic equilibrium a formal de-scription and simulation analysis studyrdquo Applied Mathe-matical Modelling vol 39 no 7 pp 1794ndash1803 2015

[5] S Ruan ldquoOn nonlinear dynamics of predator-prey modelswith discrete delayrdquo Mathematical Modelling of NaturalPhenomena vol 4 no 2 pp 140ndash188 2009

[6] M Umar Z Sabir and M A Z Raja ldquoIntelligent computingfor numerical treatment of nonlinear prey-predator modelsrdquoApplied Soft Computing vol 80 pp 506ndash524 2019

[7] V Volterra and M Brelot Leccedilons sur la theorie mathematiquede la lutte pour la vie Paris Gauthier-Villars Paris France1931

[8] M Brelot ldquoSur le probleme biologique hereditaire de deuxespeces devorante et devoreerdquo Annali di Matematica Pura edApplicata vol 9 no 1 pp 57ndash74 1931

[9] J M Cushing Integrodifferential Equations and Delay Modelsin Population Dynamics Vol 20 Springer Science amp BusinessMedia Berlin Germany 2013

[10] N Apreutesei and G Dimitriu ldquoOn a prey-predator reaction-diffusion system with holling type iii functional responserdquoJournal of Computational and Applied Mathematics vol 235no 2 pp 366ndash379 2010

[11] M R Garvie ldquoFinite-difference schemes for reac-tionndashdiffusion equations modeling predatorndashprey interac-tions in MATLABrdquo Bulletin of Mathematical Biology vol 69no 3 pp 931ndash956 2007

[12] M R Garvie and C Trenchea ldquoFinite element approximationof spatially extended predatorndashprey interactions with theholling type II functional responserdquo Numerische Mathematikvol 107 no 4 pp 641ndash667 2007

[13] S Ruan ldquoAbsolute stability conditional stability and bifur-cation in Kolmogorov-type predator-prey systems with dis-crete delaysrdquo Quarterly of Applied Mathematics vol 59 no 1pp 159ndash173 2001

[14] X-Z He ldquoStability and delays in a predator-prey systemrdquoJournal of Mathematical Analysis and Applications vol 198no 2 pp 355ndash370 1996

[15] Z Lu and W Wang ldquoGlobal stability for two-species Lotka-Volterra systems with delayrdquo Journal of MathematicalAnalysis and Applications vol 208 no 1 pp 277ndash280 1997

[16] T Faria ldquoStability and bifurcation for a delayed predator-preymodel and the effect of diffusionrdquo Journal of MathematicalAnalysis and Applications vol 254 no 2 pp 433ndash463 2001

[17] P T Young ldquoCongruences for Bernoulli euler and stirlingnumbersrdquo Journal of Number Ceory vol 78 no 2pp 204ndash227 1999

[18] F Mirzaee and S Bimesl ldquoA uniformly convergent eulermatrix method for telegraph equations having constant co-efficientsrdquo Mediterranean Journal of Mathematics vol 13no 1 pp 497ndash515 2016

[19] F Mirzaee and N Samadyar ldquoExplicit representation oforthonormal Bernoulli polynomials and its application forsolving VolterrandashFredholmndashHammerstein integral equa-tionsrdquo SeMA Journal vol 77 no 1 pp 81ndash96 2019

[20] F Mirzaee N Samadyar and S F Hoseini ldquoEuler polynomialsolutions of nonlinear stochastic Ito-Volterra integral equa-tionsrdquo Journal of Computational and Applied Mathematicsvol 330 pp 574ndash585 2018

[21] G-S Cheon ldquoA note on the Bernoulli and Euler polyno-mialsrdquo Applied Mathematics Letters vol 16 no 3 pp 365ndash368 2003

[22] X-P Yan and Y-D Chu ldquoStability and bifurcation analysisfor a delayed Lotka-Volterra predator-prey systemrdquo Journal ofComputational and Applied Mathematics vol 196 no 1pp 198ndash210 2006

[23] S Yuzbası ldquoBessel collocation approach for solving contin-uous population models for single and interacting speciesrdquoApplied Mathematical Modelling vol 36 no 8 pp 3787ndash3802 2012


prey species respectivelymdashsystem (1) reduces to the fol-lowing LotkandashVolterra predator-prey model with two dis-crete delays

p1prime(t) p1(t) r1 minus a11p1(t) minus a12p2 t minus τ1( 11138571113858 1113859

p2prime(t) p2(t) minusr2 + a21p1 t minus τ2( 1113857 minus a22p2(t)1113858 1113859

⎧⎨

⎩ (2)

We will find the approximate solutions of the system (2)with initial conditions

p1(0) α1

p2(0) α2(3)

where α1 and α2 are positive numberse exact solution for the predator-prey models in the

general form does not exist therefore numerical methodsare needed to computing the population densities of the preyand predator Somemethods [10ndash12] are proposed to handlean approximate solution of the predatorndashprey system suchas finite-difference finite element and spectral methods

e organization of this paper is as follows In Section 2we study the stability of the predator-prey model In Section3 matrix relations for Euler polynomials connected withBernoulli ones will be discussed In Section 4 we shall

present our method in detail Finally we give a few nu-merical examples in Section 5

2 Stability

In this section we review some results on the stability of thepredator-prey model To begin with we first consider system(2) appearing in the interspecific interaction terms of bothequations

p1prime(t) p1(t)f p1(t) p2 t minus τ1( 1113857( 1113857

p2prime(t) p2(t)g p1 t minus τ2( 1113857 p2(t)( 1113857

⎧⎨

⎩ (4)

where τi ge 0 i 1 2 is a constant Denote C C([minusτ

0] times R) where τ max τ1 τ21113864 1113865 Assume that f R times C⟶ R

and g R times C⟶ R satisfy the following assumptions

(S1) there exists a point (plowast1 plowast2 ) with plowast1 plowast2 gt 0 forwhich f(plowast1 plowast2 ) g(plowast1 plowast2 ) 0(S2) f and g are continuously differentiable such thatzfzp1 lt 0 zfzp2 lt 0 zgzp1 gt 0 and zgzp2 gt 0

Assumption (S1) ensures that (plowast1 plowast2 ) is a positiveequilibrium point of system (4) Let ϕ(t) p1(t) minus plowast1 andψ(t) p2(t) minus plowast2 en the linearized system at (plowast1 plowast2 ) is

ϕprime(t) zf

zp1plowast1 plowast2( 11138571113888 1113889 plowast1( 1113857ϕ(t) +

zf

zp2plowast1 plowast2( 11138571113888 1113889 plowast1( 1113857ψ t minus τ1( 1113857

ψprime(t) zg

zp1plowast1 plowast2( 11138571113888 1113889 plowast2( 1113857ϕ t minus τ2( 1113857 +

zg

zp2plowast1 plowast2( 11138571113888 1113889 plowast2( 1113857ψ(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(5)

erefore we have the following theorem

Theorem 1 Suppose that f and g satisfy the assumptions(S1) and (S2) Define

a zf


b zf


c zg


d zg


(6)

en the positive equilibrium (plowast1 plowast2 ) of the delayedpredator-prey system (4) is absolutely stable if and only ifa d + bcgt 0

For proof see [13]By eorem 1 we have the following condition If

r1a21 minus r2a11 gt 0 (7)

then system (2) has a positive equilibrium Elowast (plowast1 plowast2 )where

plowast1

r1a22 + r2a12

a11a22 + a12a21

plowast2

r1a21 minus r2a11

a11a22 + a12a21

(8)

e condition a d + bcgt 0 becomes a11a22 minus a12a21 gt 0us by eorem 1 we have the following result on thestability of Elowast (plowast1 plowast2 )

Corollary 1 If condition (7) is satisfied ie if the positiveequilibrium Elowast (plowast1 plowast2 ) of system (2) exists then it isabsolutely stable if and only if a11a22 minus a12a21 gt 0






G(x t) ≔text


n0Bn(x)

tn

n (9)


t


n0βn

tn

n (10)


1113944nminus1

k0

n

k1113888 1113889Bk(t) nt

nminus 1 n 2 3

En(t) + 1113944n

k0

n

k1113888 1113889Ek(t) 2t

n n 1 2

(11)



(2) Em(t + 1) + Em(t) 2tm mge 1


En(t) 1

n + 11113944

n+1

k12 minus 2k+1

1113872 1113873

n + 1

k


(12)


0 1 n [21]




h(t)≃ 1113957p(t) 1113944N

n0pnEn(t) PE(t)



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

where

P p0 p1 pN1113858 1113859

E(t) E0(t) E1(t) EN(t)1113858 1113859T

Eprime(t) DE(t)

(15)

such that

D




⋮ ⋮ ⋱ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


E(t) MT(t) (17)

where


M


2 minus 23( 11138572

2⎛⎝ ⎞⎠


2 minus 24( 11138573

3⎛⎝ ⎞⎠

3β3

2 minus 23( 11138573

2⎛⎝ ⎞⎠


⋮ ⋮ ⋮ ⋱ ⋮

2 minus 2N+2( 1113857N + 1

N + 1⎛⎝ ⎞⎠

N + 1βN+1

2 minus 2N+1( 1113857N + 1

N

⎛⎝ ⎞⎠

N + 1βN

2 minus 2N( 1113857N + 1



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)


[1 t t2 tN]T






1113896 (19)









Bτ(i j)

i minus 1


0 ilt j

⎧⎪⎪⎨

⎪⎪⎩(21)


AotimesB


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)






jprimeT(t)1113874 1113875




(24)

where

Pij Pi otimesPj

M MotimesM


jprime

T(t) T(t)otimesT(t)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)





⎧⎨

⎩ (26)

Suppose that



A1(t) a12MB 0 τ1( 1113857T(t)

A2(t) minusa21MB 0 τ2( 1113857T(t)

C1(t) a11MT(t)

C2(t) a22MT(t)

(27)


PM(t) + PA(t) + 1113954PC(t) f(t) (28)

where

P P1 P21113858 1113859

M(t) M1(t) 0

0 M2(t)1113890 1113891

P P12P211113960 1113961

A(t) A1(t) 0

0 A2(t)1113890 1113891

1113954P P11P221113960 1113961

C(t) C1(t) 0

0 C2(t)1113890 1113891

f(t) [0 0]1times2

(29)


ts b

Ns s 1 2 N (30)



P1MT(0) α1

P2MT(0) α2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)





1113868111386811138681113868 (32)



1113868111386811138681113868 ti i

1000 i 1 2 10001113882 1113883

(33)




11138681113868111386811138681113868

11138681113868111386811138681113868 0


11138681113868111386811138681113868

11138681113868111386811138681113868 0

⎧⎪⎨

⎪⎩

(34)





⎧⎨

⎩ (35)





tR1N(t) R2N(t)













0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)


05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05






⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References





























G(x t) ≔text


n0Bn(x)

tn

n (9)


t


n0βn

tn

n (10)


1113944nminus1

k0

n

k1113888 1113889Bk(t) nt

nminus 1 n 2 3

En(t) + 1113944n

k0

n

k1113888 1113889Ek(t) 2t

n n 1 2

(11)



(2) Em(t + 1) + Em(t) 2tm mge 1


En(t) 1

n + 11113944

n+1

k12 minus 2k+1

1113872 1113873

n + 1

k


(12)


0 1 n [21]




h(t)≃ 1113957p(t) 1113944N

n0pnEn(t) PE(t)



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

where

P p0 p1 pN1113858 1113859

E(t) E0(t) E1(t) EN(t)1113858 1113859T

Eprime(t) DE(t)

(15)

such that

D




⋮ ⋮ ⋱ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


E(t) MT(t) (17)

where


M


2 minus 23( 11138572

2⎛⎝ ⎞⎠


2 minus 24( 11138573

3⎛⎝ ⎞⎠

3β3

2 minus 23( 11138573

2⎛⎝ ⎞⎠


⋮ ⋮ ⋮ ⋱ ⋮

2 minus 2N+2( 1113857N + 1

N + 1⎛⎝ ⎞⎠

N + 1βN+1

2 minus 2N+1( 1113857N + 1

N

⎛⎝ ⎞⎠

N + 1βN

2 minus 2N( 1113857N + 1



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)


[1 t t2 tN]T






1113896 (19)









Bτ(i j)

i minus 1


0 ilt j

⎧⎪⎪⎨

⎪⎪⎩(21)


AotimesB


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)






jprimeT(t)1113874 1113875




(24)

where

Pij Pi otimesPj

M MotimesM


jprime

T(t) T(t)otimesT(t)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)





⎧⎨

⎩ (26)

Suppose that



A1(t) a12MB 0 τ1( 1113857T(t)

A2(t) minusa21MB 0 τ2( 1113857T(t)

C1(t) a11MT(t)

C2(t) a22MT(t)

(27)


PM(t) + PA(t) + 1113954PC(t) f(t) (28)

where

P P1 P21113858 1113859

M(t) M1(t) 0

0 M2(t)1113890 1113891

P P12P211113960 1113961

A(t) A1(t) 0

0 A2(t)1113890 1113891

1113954P P11P221113960 1113961

C(t) C1(t) 0

0 C2(t)1113890 1113891

f(t) [0 0]1times2

(29)


ts b

Ns s 1 2 N (30)



P1MT(0) α1

P2MT(0) α2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)





1113868111386811138681113868 (32)



1113868111386811138681113868 ti i

1000 i 1 2 10001113882 1113883

(33)




11138681113868111386811138681113868

11138681113868111386811138681113868 0


11138681113868111386811138681113868

11138681113868111386811138681113868 0

⎧⎪⎨

⎪⎩

(34)





⎧⎨

⎩ (35)





tR1N(t) R2N(t)













0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)


05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05






⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References

























M


2 minus 23( 11138572

2⎛⎝ ⎞⎠


2 minus 24( 11138573

3⎛⎝ ⎞⎠

3β3

2 minus 23( 11138573

2⎛⎝ ⎞⎠


⋮ ⋮ ⋮ ⋱ ⋮

2 minus 2N+2( 1113857N + 1

N + 1⎛⎝ ⎞⎠

N + 1βN+1

2 minus 2N+1( 1113857N + 1

N

⎛⎝ ⎞⎠

N + 1βN

2 minus 2N( 1113857N + 1



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)


[1 t t2 tN]T






1113896 (19)









Bτ(i j)

i minus 1


0 ilt j

⎧⎪⎪⎨

⎪⎪⎩(21)


AotimesB


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)






jprimeT(t)1113874 1113875




(24)

where

Pij Pi otimesPj

M MotimesM


jprime

T(t) T(t)otimesT(t)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)





⎧⎨

⎩ (26)

Suppose that



A1(t) a12MB 0 τ1( 1113857T(t)

A2(t) minusa21MB 0 τ2( 1113857T(t)

C1(t) a11MT(t)

C2(t) a22MT(t)

(27)


PM(t) + PA(t) + 1113954PC(t) f(t) (28)

where

P P1 P21113858 1113859

M(t) M1(t) 0

0 M2(t)1113890 1113891

P P12P211113960 1113961

A(t) A1(t) 0

0 A2(t)1113890 1113891

1113954P P11P221113960 1113961

C(t) C1(t) 0

0 C2(t)1113890 1113891

f(t) [0 0]1times2

(29)


ts b

Ns s 1 2 N (30)



P1MT(0) α1

P2MT(0) α2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)





1113868111386811138681113868 (32)



1113868111386811138681113868 ti i

1000 i 1 2 10001113882 1113883

(33)




11138681113868111386811138681113868

11138681113868111386811138681113868 0


11138681113868111386811138681113868

11138681113868111386811138681113868 0

⎧⎪⎨

⎪⎩

(34)





⎧⎨

⎩ (35)





tR1N(t) R2N(t)













0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)


05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05






⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References




























⎧⎨

⎩ (26)

Suppose that



A1(t) a12MB 0 τ1( 1113857T(t)

A2(t) minusa21MB 0 τ2( 1113857T(t)

C1(t) a11MT(t)

C2(t) a22MT(t)

(27)


PM(t) + PA(t) + 1113954PC(t) f(t) (28)

where

P P1 P21113858 1113859

M(t) M1(t) 0

0 M2(t)1113890 1113891

P P12P211113960 1113961

A(t) A1(t) 0

0 A2(t)1113890 1113891

1113954P P11P221113960 1113961

C(t) C1(t) 0

0 C2(t)1113890 1113891

f(t) [0 0]1times2

(29)


ts b

Ns s 1 2 N (30)



P1MT(0) α1

P2MT(0) α2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)





1113868111386811138681113868 (32)



1113868111386811138681113868 ti i

1000 i 1 2 10001113882 1113883

(33)




11138681113868111386811138681113868

11138681113868111386811138681113868 0


11138681113868111386811138681113868

11138681113868111386811138681113868 0

⎧⎪⎨

⎪⎩

(34)





⎧⎨

⎩ (35)





tR1N(t) R2N(t)













0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)


05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05






⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References


























tR1N(t) R2N(t)













0

01

02

03

04

05

06

07

08

09

1

0 50 100 150 200 250 300 350 400 450

(a)

0

01

02

03

04

05

06

07

08

09

1

100 150 200 250 300 350 4000 50 450

(b)


05 06 07 08 09 1 11 12 1302

025

03

035

04

045

05






⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References




























⎧⎨

⎩ (36)


p1(0) 13

p2(0) 06(37)





⎧⎨

⎩ (38)







6 Conclusion



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



tA 045 B 015C 045 D 09

A 08 B 04C 04 D 06

A 016 B 048C 03 D 015



Data Availability




References

























Data Availability




References

























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