Proportional population models and stability analysis on time scales

Theorem 1 ([22, 13])

Consider a differential equation, typically a nonlinear function of 𝑁,

In this case, equilibrium points, where f ⁢ ( N ) = 0 , are defined as N = N ∗ , where d ⁢ N d ⁢ t | N = N ∗ = 0 .

Theorem 2 ([20])

For systems of equations, it is possible to find equilibrium points. Let 𝐴 be a 2 × 2 constant matrix and let 𝑥 be a 2 × 1 vector. Consider d ⁢ x d ⁢ t = A ⁢ x . For this system, points where A ⁢ x = 0 are equilibrium points.

Theorem 3 ([16])

Let x = 0 be an equilibrium point for d ⁢ x d ⁢ t = f ⁢ ( x ) , where f : D → R n and D ⊂ R n is a domain including x = 0 . Let V : D → R be a continuously differentiable function where

Then x = 0 is stable (according to the Lyapunov criterion). Furthermore, if d ⁢ V d ⁢ t < 0 in D − { 0 } , then x = 0 is asymptotically stable. Here V ⁢ ( x ) is called a Lyapunov function.

Theorem 4 ([14])

Consider the behavior of solutions for the autonomous system

near an isolated critical point ( x 0 , y 0 ) , where f 1 ⁢ ( x 0 , y 0 ) = f 2 ⁢ ( x 0 , y 0 ) = 0 . A critical point is isolated if its neighborhood includes no other critical point. Here f 1 and f 2 are continuously differentiable on a neighborhood of ( x 0 , y 0 ) . Let x 0 = y 0 = 0 without loss of generality. Otherwise, we make substitutions u = x − x 0 , v = y − y 0 . Then

so (2.1) is equivalent to

that has an isolated critical point ( 0 , 0 ) .

Theorem 5 ([14])

Taylor’s formula for two variable functions implies that if f 1 ⁢ ( x , y ) is continuously differentiable near a fixed point ( x 0 , y 0 ) , then

If one applies Taylor’s formula to both f 1 and f 2 in (2.2) and supposes ( x 0 , y 0 ) is an isolated critical point,

this yields

where r ⁢ ( u , v ) and the analogous remainder term s ⁢ ( u , v ) for f 2 satisfy

Then, when 𝑢 and 𝑣 are small, r ⁢ ( u , v ) and s ⁢ ( u , v ) are small even in comparison with 𝑢 and 𝑣.

If one drops the presumably small nonlinear terms r ⁢ ( u , v ) and s ⁢ ( u , v ) in (2.3), the following linear system is obtained:

whose constant coefficients of u and v are values f 1 x ⁢ ( x 0 , y 0 ) and f 1 y ⁢ ( x 0 , y 0 ) of f 1 and f 2 at ( x 0 , y 0 ) . Because (2.3) is equivalent to the original nonlinear system u ′ = f 1 ⁢ ( x 0 + u , y 0 + v ) , v ′ = f 2 ⁢ ( x 0 + u , y 0 + v ) in (2.2), the conditions in (2.4) suggest that the linearized system in (2.5) closely approximates the given nonlinear system when ( u , v ) is close to ( 0 , 0 ) .

Supposing ( 0 , 0 ) is also an isolated critical point of the linear system, and the remainder terms in (2.3) satisfy the condition in (2.4), the original system x ′ = f 1 ⁢ ( x , y ) , y ′ = f 2 ⁢ ( x , y ) is said to be almost linear at ( x 0 , y 0 ) . So its linearization at ( x 0 , y 0 ) is a linear system in (2.5). Briefly, this linearization is the linear system u ′ = J ⁢ u (where u = [ u ⁢ v ] T ) whose coefficient matrix is the Jacobian matrix

of f 1 and f 2 , evaluated at ( x 0 , y 0 ) .

Theorem 6 ([14])

Let λ 1 and λ 2 be eigenvalues of the coefficient matrix of

for a ⁢ d − b ⁢ c ≠ 0 , associated with the almost linear system

1. If λ 1 = λ 2 are real eigenvalues, then the critical point ( 0 , 0 ) of (2.7) is either a node or a spiral point, and is asymptotically stable if λ 1 = λ 2 < 0 , unstable if λ 1 = λ 2 > 0 .

2. If λ 1 and λ 2 are pure imaginary, then ( 0 , 0 ) is either a center or a spiral point, and may be either asymptotically stable, stable, or unstable.

3. Otherwise, the critical point ( 0 , 0 ) of the almost linear system in (2.7) is of the same type and stability as the critical point ( 0 , 0 ) of the associated linear system in (2.6).

Definition 1 ([4])

Let α ∈ [ 0 , 1 ] and let k 0 , k 1 : [ 0 , 1 ] × T → [ 0 , ∞ ) be continuous functions

Definition 2 ([4])

The proportional Δ-integral of 𝑓 on [ a , b ] is

where a , b ∈ T , a < b , and k 0 ⁢ ( α , t ) − μ ⁢ ( t ) ⁢ k 1 ⁢ ( α , t ) ≠ 0 , α ∈ ( 0 , 1 ] , t ∈ T .

A function f : T → R is called a conformable regressive function if

for any α ∈ ( 0 , 1 ] and for any t ∈ T . The set of all conformable regressive functions on 𝕋 is denoted by R c .

Definition 4 ([4])

Let α ∈ ( 0 , 1 ] , p ∈ R c . For t , t 0 ∈ T , the proportional exponential function is

Theorem 7 ([4])

Let 𝑓 and 𝑔 be Δ-differentiable at t ∈ T κ . Then

1. D α ⁢ ( f + g ) ⁢ ( t ) = D α ⁢ f ⁢ ( t ) + D α ⁢ g ⁢ ( t ) ,

2. D α ⁢ ( f ⁢ g ) ⁢ ( t ) = ( D α ⁢ f ⁢ ( t ) ) ⁢ g ⁢ ( t ) + f σ ⁢ ( t ) ⁢ ( D α ⁢ g ⁢ ( t ) ) − k 1 ⁢ ( α , t ) ⁢ f σ ⁢ ( t ) ⁢ g ⁢ ( t ) .

Theorem 8 ([4])

Let u : T × T → C be Δ differentiable with respect to the first variable or the second variable at ( t , s ) ∈ T × T . The proportional Δ-partial derivatives with respect to the first variable or the second variable at ( t , s ) are

respectively.

Theorem 9 ([4])

Let h ∈ C r ⁢ d 1 ⁢ ( T ) and g ∈ R c be such that

for z ∈ H c ⁢ ( h ) , where H c ⁢ ( h ) includes all z ∈ R c for z − k 1 ∈ R c and

Let f : T → C be regulated. The proportional Laplace transform of 𝑓 on time scales is

for z ∈ D c ⁢ ( f ) , where D c ⁢ ( f ) includes z ∈ H c ⁢ ( h ) when the improper integral exists.

Theorem 10 ([4])

Let A ∈ R c and q : T → R n be rd-continuous. Let also t 0 ∈ T , y 0 ∈ R n , and let k 0 ⁢ I + μ ⁢ A be invertible on 𝕋. Then

has a unique solution

where B = ( μ ⁢ k 1 − k 0 ) ⁢ A ⁢ ( k 0 ⁢ I + μ ⁢ A ) − 1 .

Theorem 11 ([4])

Let 𝐴 be an n × n constant matrix and A ∈ R c , where R c is the set of all proportional regressive functions on 𝕋, and t 0 ∈ T . Consider the system

where x = ( x 1 ⁢ x 2 ⁢ ⋯ ⁢ x n ) T , 𝑇 is transpose. Let λ , ξ be an eigenpair of 𝐴. Then

is a solution of (2.8).

Theorem 12 (Proportional Putzer algorithm [4])

Let A ∈ R c be an n × n constant matrix and t 0 ∈ T . If λ 1 , λ 2 , … , λ n are eigenvalues of 𝐴, then

where

is solution of

The solution to the Malthusian proportional model

where β , r ∈ R .

Proof

Let us apply the proportional Laplace transform to both sides of (3.1) to get

and so

If the inverse proportional Laplace transform of both sides is taken from here, we get the solution of the Malthusian problem as

where L c − 1 is called inverse proportional Laplace transform, which satisfies the property L c − 1 ( F ) ( z ) = f ( . , 0 ) with L c ( f ( . , 0 ) ) ( z ) = F ( z ) . ∎

The only equilibrium point of equation (3.1) is N ∗ = 0 when D α ⁢ N = 0 . This equation can show both exponential growth and exponential decreasing behavior. According to the definition of the proportional exponential function E r , exponential growth occurs when r > 0 , leading to unbounded growth. In this case, solution (3.2) is unstable. Conversely, when r < 0 , N ⁢ ( t ) decreases over time and converges to zero. In this case, solution (3.2) is stable.

Consider the restricted two living state uniform growth model

The solution of this system is

where

Proof

Since

we get

Here λ 1 = 0 and λ 2 = − ( m 12 + m 21 ) . Then the eigenvector corresponding to each eigenvalue will be found. For λ 1 = 0 , we obtain

Thus, the eigenvector corresponding to the eigenvalue λ 1 is

Similarly, let us find the corresponding eigenvector λ 2 = − ( m 12 + m 21 ) . Since

we get

So the eigenvector corresponding to λ 2 is

According to Theorem 11, the solution of system (3.3) is

Let us consider system (3.3). Let us find the equilibrium points ℓ 1 ∗ and ℓ 2 ∗ so that D α ⁢ l 1 = 0 and D α ⁢ l 2 = 0 , i.e.,

When these two situations are considered together, the following is obtained:

From here,

Thus, we choose ℓ 1 ∗ = 0 and l 2 ∗ = 0 . So the equilibrium point is ( ℓ 1 ∗ , ℓ 2 ∗ ) = ( 0 , 0 ) .

Consider system (3.3). By Theorem 3, we choose

Then

Here, if m 12 , m 21 , k 1 > 0 , ℓ 1 2 + ℓ 1 ⁢ ℓ 1 σ > ℓ 1 ⁢ ℓ 2 + ℓ 1 ⁢ ℓ 2 σ , ℓ 2 2 + ℓ 2 ⁢ ℓ 2 σ > ℓ 1 ⁢ ℓ 2 + ℓ 1 σ ⁢ ℓ 2 , and ℓ 1 ⁢ ℓ 1 σ + ℓ 2 ⁢ ℓ 2 σ > 0 , we obtain

Thus, the ( ℓ 1 ∗ , ℓ 2 ∗ ) = ( 0 , 0 ) point is stable in the Lyapunov sense.

Consider the interspecific competition model

and N ⁢ ( t 0 ) = N 0 . The linearized form of the system around the equilibrium point ( N 1 ∗ , N 2 ∗ ) is given by

Then the solution of the linearized system is expressed as

where