SOP TRANSACTIONS ON STATISTICS AND ANALYSIS ISSN(Print): 2373-843X ISSN(Online): 2373-8448 DOI: 10.15764/STSA.2014.03001 Volume 1, Number 3, October 2014

SOP TRANSACTIONS ON STATISTICS AND ANALYSIS

Stochastic Dynamical Theta-Logistic Population Growth Model Morteza Khodabin*, Neda Kiaee Department of Mathematics, College of basic sciences, Karaj Branch, Islamic Azad University, Alborz, Iran. *Corresponding author: [email protected]

Abstract: In this paper, the stochastic theta-logistic population growth model is introduced. The existence, uniqueness and asymptotic stability of the solution are discussed. In order to represent the effect of theta on logistic curve and the procedure of population growth, the nonlinear stochastic differential equation of this model is solved numerically and for more clarification Irans population data in the period between 1921 to 2011 is applied. The theta-logistic is a simple and flexible model for describing how the growth rate of population increases as theta increases. Keywords: Brownian Motion Process; Itoˆ Integral; Itoˆ Process; Theta-Logistic Growth Model

1. INTRODUCTION Population growth models are abstract representation of the real world objects, systems or processes to illustrate the theoretical concepts that these days are increasingly being used in more applied situations such as predicting future outcomes or simulation experiments. In mathematical literature, many population models have been considered, from deterministic and stochastic population models where the population size is represented by a discrete random variable, to very complex continuous stochastic models. A nonrandom case, ignores natural variation and produces a single value result, whereas a stochastic model incorporates some natural variations in to model to state unpredictable situations such as weather or random fluctuations in resources and itgenerates a mean or most probable result. Nowadays, the well-known model like logistic isplaying a major role in modern ecological theory. In paper [1], the Laguerre-type derivatives and the Laguerre-type exponentials are introduced and then by using Laguerre-type exponentials the LExponential and L-Logistic population growth models are derived, and output of these models is given for world population growth in the period 1955-2005. The paper [2], develops a stochastic logistic population growth model with immigration and multiple births. The differential equation for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model is reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the normal approximation, and even more accurately using the saddle point approximation and it is shown that in comparison with the corresponding single birth model, the 1

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multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance. The paper [3], discusses the existence, uniqueness and asymptotic stability of the solution to the stochastic population model with the Allee effect. In paper [4], the stochastic and generalized stochastic exponential population growth models are introduced and the expectations and variances of solutions are obtained. So, in the present paper:After this introduction, Section 2 presents some preliminaries of stochastic calculus. In Section 3, the deterministic logistic population growth model is defined. Thereafter, section 4 introduces the stochastic population growth model. As a case study in section 5, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals along 1291-2010 years. Finally, section 6 brings about brief conclusion.

2. PRELIMINARIES Definition 2.1. (Brownian motion process). Stochastic process {B(t);t process, if it satisfies the following properties

0} is called Brownian motion

(i) B(0) = 0. (ii) B(t) has stationary increments. (iii) B(t) has independent increments. (iv) B(t) ⇠ N(0,t). Definition 2.2. Let {N(t)}t 0 be an increasing family of s -algebras of sub-sets of W. A process g(t, w) from [0, •)⇥W to Rn is called N(t)-adapted if for each t 0 the function w ! g(t, w) is N(t)-measurable, [6, p.25]. Definition 2.3. Let n = n(S, T ) be the class of functions f (t, w) : [0, •) ⇥ W ! R such that, (i) (t, w) ! f (t, w), is B ⇥ F -measurable, where B denotes the Borel s -algebra on [0, •) and F is the s -algebra on W. (ii) f (t, w) is Ft -adapted, where Ft is the s -algebra generated by the random variables B(s); s  t. ⇥R (iii) E ST f 2 (t, w)dt] < •. see [6, p.29]

Definition 2.4. (The Itˆo integral), [6, p.29]. Let f 2 n(S, T ), then the Itˆo integral of f (from S to T) is defined by Z T S

f (t, w)dB(t)(w) = lim

Z T

n!• S

fn (t, w)dB(t)(w),

(limit

in

where fn is a sequence of elementary functions such that E Theorem 2.5.

⇥Z

S

T

( f (t, w)

⇤ fn (t, w))2 dt ! 0,

as

n ! •.

(The Itˆo isometry). Let f 2 n(S, T ), then

2

⇥Z E (

S

T

⇤ ⇥Z f (t, w)dB(t)(w))2 = E

S

T

⇤ f 2 (t, w)dt .

L2 (P))

Stochastic Dynamical Theta-Logistic Population Growth Model

Proof. see [6, p.29]

Theorem 2.6.

Let f , g 2 n(0, T ) and 0  S < U < T then (i)

RT

(ii)

S

RT S

f dB(t) =

RT S

S

f dB(t) +

(c f + g)dB(t) = c RT

(iii) E[ (iv)

RU

S

f dB(t)] = 0.

RT S

RT U

f dB(t), for all w.

f dB(t) +

RT S

gdB(t), for all w.

f dB(t) is Ft -measurable.

This clearly holds for all elementary functions, so by taking limits we obtain this for all f , g 2 u(0, T ), Proof.[6,p.30].

Definition 2.7. (1-dimensional Itˆo processes), [6, p.43]. Let B(t) be 1-dimensional Brownian motion on (W, F , P). A 1-dimensional Itˆo process (stochastic integral) is a stochastic process X(t) on (W, F , P) of the form Z Z X(t) = X(0) +

t

0

t

u(s, w)ds +

0

v(s, w)dB(s),

or

dX(t) = udt + vdB(t), where

⇥Z t 2 P v (s, w)ds < •, 0

⇥Z t P | u(s, w) | ds < •, 0

f or

all

f or

all

(1) ⇤ 0 = 1,

t t

⇤ 0 = 1.

Theorem 2.8.

(The 1-dimensional Itˆo formula). Let X(t) be an Itˆo process given by (1) and g(t, x) 2 C2 ([0, •) ⇥ R), then Y (t) = g t, X(t) , is again an Itˆo process, and dY (t) =

∂g ∂g 1 ∂ 2g 2 t, X(t) dt + t, X(t) dX(t) + t, X(t) dX(t) , ∂t ∂x 2 ∂ x2

(2)

where (dX(t))2 = (dX(t))(dX(t)) is computed according to the rules dt.dt = dt.dB(t) = dB(t).dt = 0,

dB(t).dB(t) = dt.

(3)

Proof. see[6, p.44]. In the following models, we suppose that: 1. N(t) is the number of population individuals at time t, 2. N(0) is the nonrandom initial number at time t = 0, 3. a(t) is the growth rate at time t, 4. K is the environmental carrying capacity which represents the maximal population size, 5. N(t) K is the environmental resistance. 6. In deterministic cases, we assume that a(t) = r(t) is an accurate and nonrandom given function whereas

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in stochastic forms, a(t) at time t is not completely definite and it depends on some random environment effects, i.e. a(t) = r(t) + ”noise” , where r(t) is a nonrandom function of time variable that means the growth rate of population at time t whereas we do not know the exact behavior of ”noise” term, we can set, a(t) = r(t) + a(t)W (t), where W (t) = dB(t) dt is one dimensional white noise process and B(t) is a one dimensional brownian motion and a(t) is nonrandom function that shows the infirmity and intensity of noise at time t.

3. DETERMINISTIC AND STOCHASTIC DYNAMICAL THETA-LOGISTIC MODELS Consider the following dynamical logistic models dN(t) = a(t)N(t)(1 dt

N(t) ) , t K

0,

(4)

with a(t) = r(t). In dependence on the initial population size, the solution of (4) for 0 < N(t) < K and 0 < K < N(t) is obtained [5] as N(t) =

KN(0) N(0) ± (K

N(0))e

Rt

0 r(s)ds

(5)

.

If r(t) = r, (r is a constant value) in (5), we get N(t) =

KN(0) N(0) ± (K N(0))e

rt

(6)

.

Similarly, the stochastic logistic model is ⇣ dN(t) = N(t) 1 dt

N(t) ⌘⇣ dB(t) ⌘ r(t) + a(t) , t K dt

with solution [5] N(t) = where, A(t) =

Z t 0

r(s)

N(0) ± (K

0,

KN(0) , N(0)) exp[A(t)]

(7)

1 2 N(s) a (s) + a 2 (s) )ds + a(s)dB(s) . 2 K

The logistic equation can be altered in a simple manner to give the theta-logistic model. The difference in these two models lies in the factor q which controls how significant the carrying capacity term is.

4

Stochastic Dynamical Theta-Logistic Population Growth Model

3.1 Dynamical theta-logistic model Theorem 3.1.

Let q

1. Consider the following dynamical theta-logistic model ⇣ dN(t) = a(t)N(t) 1 dt

(

N(t) q ⌘ ) , K

(8)

with a(t) = r(t). Then its exact solution is given by

N(t) =

s

N q (0) + (K q

N(t) =

s

N q (0)

or

K q N q (0)

q

N q (0))e

q

Rt

q

Rt

K q N q (0)

q

(K q

N q (0))e

0

0

r(s)ds

r(s)ds

,

(9)

,

(10)

providing, 0 < N(0) < K or 0 < K < N(0).

Proof. Its obvious that N(t) = 0 and N(t) = K are the solutions to Eq. (8), so let N(t) 6= 0 and N(t) 6= K. By using ⇣ dN(t) N(t) q ⌘ = r(t)N(t) 1 ( ) , dt K

we can write,

Z t dN(s) 0

so,

Z t q 1 N (s)dN(s)

Kq

0

N(t) ln p q K q N q (t)

Then,

where C = p q

N(s)

+

N(0) K q N q (0)

N q (s)

=

Z t 0

N(0) ln p = q K q N q (0)

r(s)ds, Z t 0

r(s)ds.

Rt N(t) p = Ce 0 r(s)ds , q K q N q (t)

(11)

. In dependence on the initial population size N(0), we distinguish two cases:

i. Let 0 < N(0) < K, then Cq = (11), that

N q (0) K q N q (0)

N(t) =

ii. Let 0 < K < N(0), then Cq = from (11) that

s

K q N q (0)

q

N q (0) + (K q

N q (0) K q N q (0)

N(t) =

> 0 a.s. So that 0 < N q (t) < K q . Hence, we conclude from

s

N q (0))e

R . q 0t r(s)ds

< 0 a.s. So that 0 < K q < N q (t) , t K q N q (0)

q

N q (0)

(K q

N q (0))e

R . q 0t r(s)ds

(12)

0. Hence, we conclude (13)

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If we put q = 1 in (12) and (13), we get N(t) =

KN(0) N(0) ± (K

N(0))e

Rt

0 r(s)ds

,

which is equal to (5) and if r(t) = r, we get N(t) =

KN(0) N(0) + (K N(0))e

rt

,

which is equal to (6).

3.2 Dynamical stochastic theta-logistic model We assume that a(t) at time t is not completely definite and it depends on some random environment effects, then the stochastic model of theta-logistic population growth is given by

or

(

⇣ dN(t) = N(t) 1 dt

(

⇣ dN(t) = N(t) 1

q ( N(t) K )

N(0) = N0 ,

N(t) q ⌘⇣ dB(t) ⌘ ) r(t) + a(t) , K dt ⌘⇣ ⌘ r(t)dt + a(t)dB(t) ,

(14)

where B = {B(t),t 0}is a one-dimensional standard Brownian motion defined on a probability space {W, z, {zt }t 0 , P} with a filtration {zt }t 0 satisfying the usual conditions (it is right continuous and increasing, while z0 contains all P-null sets), N(0) is a random variable independent of B such that 0 < N(0) < K a.s., and N(t) is an unknown stochastic process, that is, the solution to Eq.(14) satisfying the initial condition N(0). Because of the logical requirement that N(t) must be positive, we cannot directly apply the usual existence-and-uniqueness theorem. That is why we need a procedure which is explained in detail in the next section.

3.3 Existence and uniqueness of the positive solution Theorem 3.2.

Let q

1. Consider the following dynamical stochastic theta-logistic model ⇣ dN(t) = N(t) 1

(

⌘ N(t) q ⌘⇣ ) r(t)dt + a(t)dB(t) . K

Then its exact solution is given by

N(t) =

s

K q N q (0) , N q (0) + (K q N q (0))exp[F(t)]

N(t) =

s

N q (0)

or

6

q

q

K q N q (0) , (K q N q (0))exp[F(t)]

Stochastic Dynamical Theta-Logistic Population Growth Model

where, F(t) =

Z t ⇣ 0

⌘ 1 2 N(s) q a (s) + a 2 (s)( ) )ds + a(s)dB(s) , 2 K

q (r(s)

providing, 0 < N(0) < K or 0 < K < N(0).

Proof. It’s obvious that N(t) = 0 and N(t) = K are the solutions to Eq. (14). So suppose that N(t) 6= 0 and N(t) 6= K, by applying the Itˆo formula we observe that d ln |

Kq

⇣ q 2Nq

N q (t) | = d ln |K q N q (t)

N q (t)|

2 (t)K q

q N q 2 (t)K q + q N 2q 2(K q N q (t))2

Thus, we obtain |

Kq

⇣ q N q 1 (t) ⌘ dN(t) K q N q (t) ⇣ q N q 1 (t) ⌘ 2 (t) ⌘ (dN(t))2 dN(t)+ N q (t) ⇣ q ⌘ (dN(t))2 . 2N 2 (t) d ln |N q (t)| =

N q (t) | = Cexp[F(t)] , N q (t)

where C=

Kq

(15)

N q (0) . N q (0)

In dependence on the initial population size, we consider two cases: i. Let 0 < N(0) < K, then C =

K q N q (0) N q (0)

> 0. a.s. So that 0 < N q (t) < K q .

Hence the solution is N(t) =

ii. Let 0 < K < N(0), then C =

s q

K q N q (0) , N q (0) + (K q N q (0))exp[F(t)]

K q N q (0) N q (0)

N(t) =

s q

(17)

< 0 a.s. So that 0 < K q < N q (t).

Hence the solution is

(16)

N q (0)

K q N q (0) , (K q N q (0))exp[F(t)]

(18)

(19)

In special cases, • If q = 1, we get dynamical stochastic logistic model. • If a(t) = 0, we get dynamical deterministic theta-logistic model. • If q = 1 and a(t) = 0, we get dynamical deterministic logistic model.

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In what follows, we formulate the existence and uniqueness theorem of the positive solution N(t) for Eq.(14). Moreover, we prove the uniform continuity of N(t) in the sense that it is continuous and its almost every sample path is uniformly continuous on t 0. In order to prove the uniform continuity of the positive solution, we apply the Kolmogorov-Centsov theorem on the continuity of a stochastic process derived from the moment property [3]. Theorem 3.3.

For any initial value N(0) such that 0 < N(0) < K, there exists a unique, uniformly continuous positive solution for Eq(14).

Proof. Let 0 < N(0) < K. In view of (16), we define a stochastic process {x(t),t x(t) := ln

Kq

N q (t) N q (t)

0} by

.

If we apply the Itˆo formula for x(t) ,we find that dx(t) = f (x(t))dt + g(x(t))dB(t) , t

0,

(20) K q N q (0)

(1 ex )

where, f (x(t)) = q r(t) + a 2 (t) 2(1+ex ) and g(x(t)) = q a(t) for all t > 0 and x(0) := ln N q (0) . It is easy to see that the functions f and g are bounded and continuous, as well as that they satisfy the local Lipschitz condition and the linear growth condition. Hence, Eq. (14) has a unique continuous solution x(t), t 0 satisfying the initial condition x(0). Since Kq N q (t) = , 1 + ex(t) so N(t) =

s q

Kq . 1 + ex(t)

We will prove that N(t) is the solution to Eq.(14). Indeed,

dN(t) = d

s ⇥q

⇤ Kq ⇤ K q ex(t) ⇥ 1 ex(t) = dx(t) + dx(t)dx(t) x(t) x(t) 2 1+e (1 + e ) 2(1 + ex(t) ) K ex (t) = p r(t)dt + a(t)dB(t) q x 1 + e (t) 1 + ex (t) ⇣ N(t) q ⌘ = N(t) 1 ( ) r(t)dt + a(t)dB(t) K

In order to prove that almost every sample path of N(t) is uniformly continuous for t consider Eq. (14) in its integral form, that is, N(t) = N(0) +

Z t 0

f (N(s))ds +

Z t 0

g(N(s))dB(s),t

where 0 < N(0) < K and f (N(s)) = r(s) N(s) 1 8

(

N(s) q ) , K

0,

0, we will

Stochastic Dynamical Theta-Logistic Population Growth Model

and

N(s) q ) . K Let 0 < u < n < •, n u  1 and p > 2. By using the H¨older inequality and the well-known moment inequality for the Itˆo integrals, we get g(N(s)) = a(s) N(s) 1

N(s)| p  2 p 1 (n

E|N(t)

+2 p 1 (

p(p 1) p ) 2 (n 2

(

u) p p

u) 2

1 1

Z t 0 n

Z

u

E[ f (Ns )] p ds E[g(Ns )] p ds.

(21)

Since, E| f (Ns )| p  (r(s) K) p , and E|g(Ns )| p  (a(s) K) p , we find from (17) that E|N(t)

p

N(s)| p  A(n

u) 2 ,

where

p(p 1) p 2 ) 2 a (s)) . 2 The application of the Kolmogorov-Centsov theorem on the continuity of a stochastic process [3], implies that almost every sample path of N(t) is locally but uniformly H¨older-continuous with exponent g 2 (0, p2p2 ) and, therefore, uniformly continuous on t 0. A = 2P 1 K p (r p (s) +

As usual, we prove the uniqueness of the positive solution by contradiction. Suppose that N1 (t) and N2 (t) are two positive solutions of Eq.(14) with the same initial value N(0), where 0 < N(0) < K. Then, d(N2 (t)

N1 (t)) = (N2 (t)

Let us denote that J(t) = N2 (t) J(t) =

N2 (t)

N1 (t))(1

N1 (t) K

)(r(t)dt + a(t)dB(t)).

N1 (t). Then,

Z t 0

J(s)(1

N2 (s)

N1 (s) K

)(r(s)ds + a(s)dB(s)).

By using the elementary inequality (a + b)2  2a2 + 2b2 , H¨older inequality, Itˆo isometry and the fact that 0 < Ni (t) < K, i = 1, 2. it follows from (16) and (18) that E(J(t))2  2(r2 (t) t + a 2 (t))E

Z t 0

(J(s))2 (1

N2 (s)

N1 (s) K

 2(r2 (t) t + a 2 (t))E

Z t 0

)2 ds

(J(s))2 ds.

Finally, the application of the Gronwall-Bellman Lemma yields that E(J(s))2 = 0 and, therefore, E(N2 (t)

N1 (t))2 = 0.

According to the Chebyshev inequality, we find for an arbitrary e > 0 P{|N2 (t) Hence, N1 (t) = N2 (t) a.s. for all t

N1 (t)|

e} 

1 E|N2 (t) e2

N1 (t)|2 = 0.

0. Thus, the proof becomes complete .

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3.4 Stability of the positive solution of the theta-logistic model Since Eq.(14) is not explicitly solvable, it is important to investigate the behavior of the positive solution in long period of time. To do this, we will apply several times the following elementary inequality, 1 ln a  a a

ln b 1  , 0 < b < a, b b

(22)

as well as the well-Known assertion: Lemma 3.1. [3]. Let f : [0, •] ! [0, •] be an integrable and uniformly continuous function. Then, lim f (t) = 0.

t!•

The following theorem describes the asymptotic mean square stability of the positive solution to Eq.(14) when the initial population size is 0 < N(0) < K. Theorem 3.4.

Let N(t) be a uniformly continuous positive solution to Eq.(14) with the initial value N(0), where 0 < N q (0) < K q . Then: (a) If r >

a2 2 (1 + q ),

then limt!• E(K q

N q (t))2 = 0.

(b) If r < a 2 q , then limt!• E(N q (t))2 = 0.

Proof. First, it follows from (16) that 0 < N q (t) < K q ; t (a) The application of the Itˆo formula to V 2 (t), where V (t) = ln K q

0, since 0 < N q (0) < K q .

ln N q (t) , t

0,

(23)

is the Lyapunov function, implies that dV 2 (t) = 2V (t)dV (t) + (dV (t))2 . consequently, we can write EV 2 (t) = EV 2 (0) + E

Z t⇥ 0

2q ln K q

ln(N q (s))

By applying the inequality (22) and the assumption r > dEV 2 (t) < dt

2q E (K q Kq =

N q (t))2 [

q (2r K 2q

2 (r K2

we get

1 (r N q (t)

Kq

(1 + q )a 2 )E(K q

N q (t) 2 a ) 2K q N q (t))2

N q (t))2 ,

a 2 ) is a positive constant. Thus, EV 2 (t) is decreasing and hence EV 2 (t) < EV 2 (0)

10

N q (s) K q N q (s) 2 (r a ) Kq 2K q (K q N q (s))2 2 2 ⇤ + q a ds. K 2q

a2 2 (1 + q ),

< M E(K q where M =

Kq

M

Z t 0

E(K q

N q (s))2 ds.

q a 2] 2K q

Stochastic Dynamical Theta-Logistic Population Growth Model

Since

EV 2 (0) = E(ln K q

we have EV 2 (0)

M

Z t 0

E(K q

ln N q (0))2 < •,

N q (s))2 ds < EV 2 (0) < •,

which implies that E(K q N q (t))2 2 L1 [0, •]. The application of Theorem 3.3 and Lemma 3.1 yields that E(K q N q (t))2 is uniformly continuous on [0, •] and lim E(K q

t!•

N q (t))2 = 0.

(b) Analogously to the first part of the proof, if we apply the Itˆo formula to V 2 (t), where V (t) = ln K q

ln(K q

N q (t))

,

t

0,

(24)

is the Lyapunov function, we see that dV 2 (t) = 2q ln(K q

ln(K q

N q (t))

+ Similarly to the previous discussion

where Z =

2 (r + q a 2 ) K2

⇤ N q (t) ⇥ K q (q 1) + N q (t) 2 (r + a )dt + a dB(t) q q K 2K

N 2q (t) 2 2 a q dt. K 2q

dEV 2 (t) 2 < Z E N q (t) , dt

is a constant, and hence EV 2 (t) is decreasing. So, EV 2 (t) + Z

since EV 2 (0) = E(ln K q ln(K q and Lemma 3.1, it follows that

Z t 0

E(N q (s))2 ds < EV 2 (0) < •,

N q (0)))2 < •. Therefore, E(N q (t))2 2 L1 [0, •]. According to Theorem 3.3 lim E N q (t)

t!•

2

= 0,

which completes the proof.

Conditions under which the positive solution to Eq.(14) is asymptotically stable in mean are given in the next theorem. Note that the greater interval is obtained for r, than the one from Theorem 3.4. Theorem 3.5.

Let N(t) be a uniformly continuous positive solution to Eq.(14) with the initial value N(0), where 0 < N(0) < K. Then: (a) If r > (b) If r


a2 2 ,

Kq

N q (t))2 a 2 dt = q

E

Z t 0

q

Kq

N q (t) [(r Kq

Kq

N q (s) (r Kq

Kq

N q (t) 2 a )dt + a dB(t)]. 2K q

(25)

N q (s) 2 a )ds. 2K q

then dEV (t) < DE(K q dt

N q (t)),

1 (2r a 2 ) is a positive constant. Likewise, since EV (t) is decreasing and EV (0) = E(lnK q where D = 2K ln(N q (0)) < •, then

EV (t) + D so that E(K q

Z t 0

E(K q

N q (s))ds < EV (0) < •

N q (t)) 2 L1 [0, •], which leads to the conclusion that lim E(K q

t!•

N q (t)) = 0. 2

(b) To prove the second part, we use the fact that r < q a2 and the Lyapunov function V (t) given by (24). Then, by applying the Itˆo formula dV (t) =

q N q 1 (t) q Nq dN(t) + K q N q (t)

2 (t)K q (q

2(K q

1) + q N 2q N q (t))2

2 (t)

(dN(t))2

⇤ N q (t) ⇥ K q (q 1) + N q (t) 2 (r + a )dt + a dB(t) . q K 2K q By repeating completely the previous procedure, we see that =q

EV (t) = EV (0)

E

Z t 0

q

N q (s) ( r Kq

K q (q

1) + N q (s) 2 a ds 2K q

dEV (t) q q < q (r + a 2 )E(N q (t)) dt 2 K = W E(N q (t)), where, W =

q 2K q

(2r + q a 2 ) > 0 is a constant. By omitting details, we conclude that lim E(N q (t)) = 0.

t!•

12

We can prove the above theorems for 0 < K < N(0) similarly. The previous theorem shows that under certain conditions the deterministic theta-logistic population model (8) and the corresponding stochastic theta-logistic differential equation (14) have a similar property-the global stability of the positive and bounded solutions.

Stochastic Dynamical Theta-Logistic Population Growth Model

4. NUMERICALLY THE SOLUTION q In this section, we solve nonlinear equation(14) for K= 160000000, r(t) = t N(t) 1 for each census N(0 period, q = 1, 2, 3, 25, 35, 50 and a = 0.05 for Iran’s population in period of 1921-2011. We see the results in the below tables.

Table 1:Exact population, predicted values and relative errors of Iran population in period of 1921-2011,q = 1, 2.

Year Population

q =1

Relative error

q =2

Relative error

1300 1305 1310 1315 1320 1325 1330 1335 1345 1355 1365 1370 1375 1385 1390

9707000 10456000 11185000 11964000 12833000 14159000 16237000 18954704 25788722 33708744 49445010 55837163 60055488 70495782 75354709

9707000 10320000 11068000 12558000 13954000 12886000 16107000 18159400 26306000 31525000 47583000 59654000 62105000 70246000 74792000

0 0.013 0.011 0.051 0.087 0.081 0.008 0.019 0.021 0.064 0.037 0.068 0.034 0.003 0.007

9710000 10350000 11150000 12750000 14280000 13090000 16670000 19520000 28870000 35570000 58410000 77140000 81030000 93940000 101050000

0.0003 0.0101 0.0031 0.0657 0.0112 0.0751 0.0261 0.0291 0.0121 0.0552 0.1811 0.3811 0.3049 0.3032 0.3401

Mean

-

-

0.0336

-

0.1198

Table 2:Exact population, predicted values and relative errors of Iran population in period of 1921-2011, q = 3, 20, 25.

q =3

Relative error

q = 20

Relative error

q = 25

Relative error

9710000 10350000 11150000 12760000 14300000 13100000 16720000 19600000 29190000 36180000 61150000 83090000 87740000 103520000 112270000

0.0003 0.0101 0.0031 0.0665 0.1141 0.074 0.029 0.034 0.103 0.055 0.237 0.489 0.461 0.473 0.491

9710000 10350000 11150000 12770000 14310000 13100000 16720000 19610000 29240000 36280000 62170000 86940000 92260000 112640000 125690000

0.0003 0.0101 0.0031 0.0673 0.115 0.074 0.029 0.034 0.133 0.0762 0.258 0.557 0.5362 0.597 0.668

9710000 10350000 11150000 12770000 14310000 13100000 16720000 19610000 29240000 36280000 62170000 86940000 92260000 112630000 125640000

0.0003 0.0101 0.003 0.067 0.115 0.074 0.029 0.034 0.133 0.076 0.258 0.557 0.536 0.597 0.833

-

0.1759

-

0.2105

-

0.2214 13

SOP TRANSACTIONS ON STATISTICS AND ANALYSIS

Theta=1

7

12

x 10

Theta=3

7

16

x 10

14 10 12 8

Y(t)

N(t)

10 6

8 6

4 4 2 2 0 1300

1320

1340

1360 t

1380

1400

0 1300

1420

1320

1340

1360 t

1380

1400

1420

1380

1400

1420

Figure 1. q = 1 and q = 3 Theta=35

7

x 10

14

14

12

12

10

10

8

6

4

4

2

2

1320

1340

1360 t

1380

1400

1420

x 10

8

6

0 1300

Theta=50

7

16

Y(t)

N(t)

16

0 1300

1320

1340

1360 t

Figure 2. q = 25 and q = 50 Table 3:Exact population, predicted values and relative errors of Iran population in period of 1921-2011, q = 35, 50, 100, ....

q = 35

14

Relative error q = 50, 100, ... Relative error

9710000 10350000 11150000 12770000 14310000 13100000 16720000 19610000 29240000 36280000 62170000 86940000 92260000 112630000 125600000

0.0003 0.0101 0.003 0.067 0.115 0.074 0.029 0.034 0.133 0.076 0.258 0.557 0.536 0.597 0.833

9710000 10350000 11150000 12770000 14310000 13100000 16720000 19610000 29240000 36280000 62170000 86940000 92260000 112630000 125590000

0.0003 0.0101 0.003 0.067 0.115 0.074 0.029 0.034 0.133 0.076 0.258 0.557 0.536 0.597 0.66,...

-

0.2214

-

0.2096

Stochastic Dynamical Theta-Logistic Population Growth Model

5. CONCLUSION The modeling of systems by deterministic differential equations usually requires the parameters that are completely known. But in some cases their values may depend on the microscopic properties of the medium in a complicated way and may fluctuate due to some external or internal random ”noise”. Here, we defined stochastic theta- logistic population growth model where the so-called parameter, population growth rate is not completely definite and it depends on some random environmental effects. Then we discussed the existence, uniqueness and asymptotic stability of the solution. Since it is impossible to find the exact solution of nonlinear stochastic differential equation of (14), we solved it numerically. In this model with the assumption of 0 < N(t) < K and q >> 1, the fraction of 1 than one, which with the increase of q get closer to one.

q ( N(t) K ) is lower

The number of population grow rapidly, while the environmental resistance effects such as flood, earthquake, infectious disease and etc. would diminish. As it’s shown from the results, q = 1 has the best relative error between the other values of q in two interpolated real data and Euler numerical method graphs. In fact the increase of theta is equal to higher relative error. Which in this pilot, the error for higher than q = 35, will be the same and population approach to maximal population size or equilibrium value rapidly with more intensity.

References [1] S. De Andreis, P. E. Ricci,(2005, December), Modelling Population Growth via LaguerreType Exponentials, Mathematical and Computer Modelling,42(13), pp. 1421-1428. Available:http://dl.acm.org/citation.cfm?id=2282157. [2] James H. Matisa, Thomas R. Kiffe,( 2004, Feb), On stochastic logistic population growth models with immigration and multiple births, Theoretical Population Biology,65(1), pp.89104.Available:http://www.ncbi.nlm.nih.gov/pubmed/14642347. [3] M. Krsti, M. Jovanovi,(2010, July), On stochastic population model with the Allee effect, Mathematical and Computer Modeling, 52(1-2) pp.370379.Available:http://www.sciencedirect.com/science/article/pii/S0895717710001214. [4] M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri,(March, 2012), Interpolation solution in generalized stochastic exponential population growth model , Applied Mathematical Modelling,36(3), pp. 10231033.Available:http://www.sciencedirect.com/science/article/pii/S0307904X11004598. [5] M. Khodabin, N. Kiaee,(August,2011), Stochastic dynamical logistic population growth model , Journal of Mathematical Sciences: Advances and Applications, 11(1) ,pp. 1129.Available:http://scientificadvances.co.in/admin/img data/485/images/[2] JMSAA 200118008 Morteza Khodabin and Neda Kiaee[11 29].pdf [6] B. Oksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Springer-Verlag, New York, (1998).

15

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