In addition, we show the stability condition of the extinction equilibrium of this model in Liapunov's sense. I also thank him for being there for me from the beginning of the work to the end. We define the boundary of a subset S of X as the set of points x∈ and becomes the closure of S.
Now from (1.4) we have f : K → Rm, K ⊂ Rm, we assume that f is continuously differentiable in some equilibrium point xe ∈ Rm. The Jacobian at an equilibrium xe, Df(xe), is essential for the stability properties of the equilibrium xe. We use these results to study the long-term behavior of the linear autonomous models (1.3), in chapter 4, after [3] and [9].
Next, we show the existence of the global vanishing of (1.10) under the assumption that P(x) ≤ P(0) for all x∈Rm+. We conclude our thesis by providing the stability condition of a hyperbolic vanishing equilibrium of (1.10).
Basic definitions and notations
A set {x:x ∈Rm} is said to be bounded, i.e. closed, if every sequence in the set is bounded and respectively the set contains all its boundary points. 2. Unless otherwise stated, in this thesis every matrix is considered to be a real-valued square matrix of order m×m. By the Bolzano-Weierstrass theorem, as we proved in Lemma 2.1.1, every sequence in a set{x∈Rm: kxkp= 1}has a subsequence that converges to a point in the set.
The eigenvaluesλ∈Cof P are the solutions of the characteristic equation of P or, equivalently, the roots of the characteristic polynomial ofP. λs) is the set of different eigenvalues of P, which is called the spectrum of P. We define the spectral radius of P as follows. The exponent mi, corresponding to each eigenvalue λi, is called the algebraic multiplicity of λi and dim(N(P −λiI)) is called the geometric multiplicity of λi. Then v is called an eigenvector of P corresponding to the eigenvalue λ. Sometimes v is also called a right eigenvector of P.
If a vector 0 6=vi is found as a solution to the equation (P −λiI)tvi = 0 for some 1 < t ≤ mi and (P −λiI)t−1vi 6= 0, then we are called an associated or generalized eigenvector corresponding to to the eigenvalueλi.
Similarity
Let P1 and P2 be similar matrices and let λ ∈ C, then there exists a non-singular matrix Q such that. If P1 and P2 are similar matrices, then λ ∈ σ(P1) is a simple eigenvalue if and only if λ∈σ(P2) is a simple eigenvalue. −1 for every t∈I[1,+∞).Then.
Jordan forms
Therefore, the matrixQi for each 1≤i≤k(and thus Q) is the matrix whose columns are the eigenvectors and the generalized eigenvectors of P. The largest Jordan blockBi(λj) inJ(λj) in (2.15) is anrj×rj matrix whererj = index(λj). Furthermore, index(λj) = 1 if and only if each Jordan blockBi(λj) is 1×1 which happens if and only if the number of eigenvectors associated with λj inQ such that Q−1P Q=J is the same as the number Jordan blocksBi(λj). Since eachJ(λi) consists ofdj = dim(N(P−λjI), we have "the algebraic multiple is equal to the geometric multiplej" This is just another way of saying that algebraic multiple and geometric multiple of λj are the same , which is the definition of λj which is semisimple.Let λ∈σ(P).Then every Jordan block associated with λ is a 1×1 matrix if and only if λ is semisimple.
Furthermore, if ρ(P) = 1, then P is convergent to G, where G is the projector of N(I−P) along R(I−P) if and only if ρ(P) is a semisimple eigenvalue of P and ρ(P) is the only eigenvalue in the unit circle. Now suppose that ρ(P) = 1 and that it is a semisimple eigenvalue and that ρ(P) is the only eigenvalue on the unit circle. Furthermore, P is Ces`aro summable to G, where G is the projector of N(I −P) along R(I−P) if and only if ρ(P) = 1, where every eigenvalue on the unit circle is semisimple .
This is equivalent to saying that every Jordaan block Bi(λj) in J is Ces`aro-summable. Consequently, δ(λj, t) becomes unbounded as→ ∞. In other words, it is necessary thatρ(P)≤1 for P to be Ces`aro-summable. From Theorem 2.3.7 and Lemma 2.3.8 we already know that P is convergent and therefore Ces`aro sums to 0 whenρ(P)<1, therefore we only need to consider the case when P has eigenvalues on the unit circle.
In other words, P cannot be Ces`ar summable if there are eigenvalues λj 6= 1 on the unit circle, so that λj is not semisimple. Similarly, if λj = 1 is not semisimple, then P cannot be Ces`aro summable because every entry on the first superdiagonal of. So, if P is Ces`aro summable and has eigenvalues λj such that |λj| = 1, then λj must be semisimple.
On the other hand, if ρ(P) = 1 and every eigenvalue on the unit circle is semisimple, then P is Ces`aro summable. This follows because every Jordan block associated with an eigenvalue λj ∈ σ(P) from Proposition 2.3.7 such that |λj|< 1 is convergent and therefore, by Lemma 2.3.8, Ces`aro can be summed to 0. For semisimple eigenvalues λj such that |λj| = 1, the associated Jordan blocks are 1×1 and therefore Ces`aro is summable because (2.29) implies.
3.4) Since ω ∈ σ(Pt), the left side is singular, so at least one of the factors on the right must be singular.
Positive matrices
Ifv >0 is the Perron vector of P satisfying v=P v, then such a vector is a positive vector satisfying kvk1 = 1 and. Thus, by Theorem 3.2.2, in addition to the Perron eigenpair (r, v), there exists a corresponding Perron vector for PT satisfying PTω=rω. There are no other nonnegative eigenvectors for P except the Perron vector v and its positive multiples.
Further non-negative matrices
Irreducible matrices
From Corollary 2.3.4 it follows that r ∈ σ(P) is a simple eigenvalue, otherwise µ = ρ(f(P)) is not a simple eigenvalue, which is impossible because f(P) >0. Seeing that P has a positive eigenvector associated with r, we recall from Theorem 3.3.1 that there exists a non-negative eigenvector x ≥ 0 associated with r. From Corollary 2.3.6, (r, x) means an eigenpair of P that (µ, x ) is an eigenpair off(P). Corollary 3.2.6 ensures that x must be a positive multiple of Perron vector off(P), and therefore x must indeed be positive. It follows from Theorem 3.4.4 that dim (N(P−I)) = 1. The uniquely defined Perron vectors for P and PT respectively are given by v= kxkx. If P ≥0 is an irreducible matrix with >1 eigenvalues on the unit circle, then P is imprimitive.
No rotation smaller than 2πh can keep ρ(P) unchanged, because Theorem 3.4.11 makes it clear that the eigenvalues on the unit circle will not return to themselves for rotation smaller than . However, Theorem 3.4.11 states that every eigenvalue on the unit circle is prime, so Theorem 2.3.9 can be applied to Pr to conclude that Pr is compact in G, that is. From theorem 3.4.8, the spectral projector onN(P−rI) alongR(P−rI) is given by G= hω,vivωT >0 kuv and ω are the corresponding Perron vectors for P and PT.
We recall the definition of instability which says that the extinction equilibrium is unstable if there exists a > 0 such that for any δ > 0 there is x0 ∈Rm such that kx0k < δ and kx(t)k ≥ for some t ∈ [0 ,∞). If there is a t ∈ I[0,+∞) for which kx(t)k > then the extinction equilibrium is unstable according to the definition of instability. A matrixB is said to be positive definite and positive semidefinite respectively, if V(x) is positive definite and positive semidefinite respectively. It says that a true symmetric matrix B = [at]1≤i,j≤m is positive definite if and only if the determinants of its governing principle minors are positive, that is, if and only if.
Let B be a positive definite matrix and let V(y) = hy, Byi be a continuous function defined for all vectors y ∈ Rm. If C is a positive-definite symmetric matrix such that (5.10) has a solution B that is also symmetric and positive-definite, then ∆V < 0. We can consider V as a Liapunov function of (5.3) (because V is continuous for ∈ Rm and ∆V(y) < 0 for ally ∈Rm and P y∈Rm.) Therefore, by Theorem 5.1.1, it is equilibrium of (5.3) is asymptotically stable. On the other hand, if the equilibrium of (5.3) is asymptotically stable, then for every positive definite symmetric matrix C, (5.10) has a unique solution B that is also symmetric and positive definite, as shown by the following theorem.
If the equilibrium of (5.3) is asymptotically stable, Equation (5.10) for every positive definite symmetric matrix C has a unique solution B which is also symmetric and positive definite. Furthermore, since C is a positive definite matrix satisfying hx, Cxi>0 forx6= 0, then forx6= 0 has hx, Bxi=. If ρ(P) > 1, then there exists a real symmetric matrix B that is not positive semidefinite such that (5.10) holds for some symmetric positive definite matrix C.
From Theorem 5.1.6, since ˜C11 is a positive definite symmetric matrix, there exists a positive definite symmetric B0 such that.
Long time behaviour of population models
Denote p(t) =kx(t)k1 for the total population at time. We make a biologically meaningful assumption that the initial density of a population x0 ≥0. To fully understand the evolution of such models, we study the theory of general matrices from the point of view of spectral theory and provide an overview of their properties. We thus study the Jordan forms of general matrices and use them to show the limit behavior of a matrix (Theorem 2.3.7 and Theorem 2.3.9).
We study the Perron-Frobenius type theorems for both positive and irreducible matrices regarding their spectral properties and give details of their proofs. We use these theorems to investigate the asymptotic behavior of linear autonomous models arising in structured population following [3]. They consider diagonalizable matrices and find that while the total population may increase or decrease depending on the spectral radius of the associated matrix, there is a proportion of individuals in each class that stabilizes as time increases.
In this thesis, we generalize their work to any matrix, whether diagonalizable or not. They find that the stability conditions of such an equilibrium are determined by the spectral radius of the associated nonlinear matrix. We assume that the associated nonlinear matrix of this model depends explicitly on the population density and that this matrix does not increase at all density levels.
