Where the work of others has been used, this is duly acknowledged in the text. Moreover, we check(t)−U(t)k=O(k) for allt∈D1, then we say that U(t) is an asymptotic approximation of u(t) in D1to within the accuracy of order k.
Outline of the Thesis
We discuss further possibilities of generalizing the model and compare our work with the existing literature. The algebraic multiplicity λ is the number of repetitions as a root of the characteristic polynomial.
Matrices with Non-negative Elements
Normed Vector Spaces
The set of all semilinear shapes on X is called a vector space, called the adjoint space of As with linear forms, for each scalarαk there is a f ∈ X∗such that (f,xk) =αk.
Projections
Each f ∈ X∗ can be expressed in a unique way as a linear combination of ej, according to
Calculus in a Banach Space
Integration in One Variable
The Derivative as a Linear Map
Properties of the Derivative
Operator Semigroups
The initial value problem (2.3) is called the abstract Cauchy problem associated with (A,D(A)) and the initial value x. An important point is that classical solutions exist if and only if the initial value x belongs to D(A).
Introduction
The Continuous-time Model of Sharpe and Lotka
The McKendrick–Von Foerster Model
Solution of the McKendrick–Von Foerster Model
Here0,t0 are respectively the initial age of an individual at time = 0 in the original population and the time of birth of an individual. Equation (3.7) that applies next to each characteristic has a different solution according to ofa>tora
Characteristic Equation
This characteristic equation has some special spectral properties that we will discuss in the following theorems. By comparing real and imaginary parts of both sides of the above equation, we have,.
Remark
Multiregional Demography
The Irreducible Migration Matrix
The reason for getting zero on the left side is that the sum of the entries in each column of C is zero. Since the sum of the elements of a column of C is zero, it follows that1 is a left eigenvector corresponding to the zero eigenvalue of Can and this.
Well-Posedness of the Model
In the following series of lemmas, we will prove the second condition of Theorem 2.18, and this will then prove our main Theorem 4.5. Since we can identify φasω(as an element iL1) by modifying the values in the null set, so we can say that φ ∈D(A) andAφ =v. Now since continuous functions with compact support are dense in L1 ([59, Theorem 2.4]), proving the fact (4.19) is a simple consequence of approximating integrable functions by continuous functions with compact support.
The specific spectral properties of the migration matrix discussed in this chapter will be used in the asymptotic analysis of the model in the next chapter. Also, the existence of a C0-semigroup (Theorem 4.5) is one of the key results needed for all our subsequent analyses. ASYMPTOTIC ANALYSIS OF A DISTURBED MODEL 39 where β∗ :=1·Bk is the 'total fertility' should provide an approximate description of the average population.
The analysis is included because of the initial and boundary conditions which are inconsistent with those of the summarized model. This makes the problem particularly turbulent and thus requires a careful analysis of boundary, corner and initial layer phenomena.
Formal Asymptotic Expansion
- Spectral Projections
- Projected System of Equations
- Bulk Approximation
- Initial Layer
- Boundary Layer
- Corner Layer
To correct the situation we need to make corrections that will take care of transients that occur near total = 0 and limit = 0. Thanks to the linearity of the problem, we approximate ˜ the solution as the sum of the bulk obtained above and the initial layer which we construct below. Since ˜v0 is a layer term, so it should only be significant in a small vicinity of the initial point=0.
Again, assuming all terms are sufficiently smooth and using the linearity of the problem, we have the following set of error equations. Again, the linearity allows us to approximate the solution by the sum of the bulk and initial layer parts, obtained above, and the boundary layer. ASYMPTOTIC ANALYSIS OF THE PERTURBED MODEL 47 as before into the projected equations, we get respectively
Since ˆv0 is a layer term, it should only be significant in a small vicinity of the initial point=0. 5.30). ASYMPTOTIC ANALYSIS OF THE DISTURBED MODEL 51 The solution of the above system can be presented as.
Model with General Mortality and Birth Matrices
The perturbed model considered in Chapters 4 and 5 deals with the impassable migration matrix, which essentially shows that every geographic part is accessible from every other. In many cases, it is found that some patches are either isolated or accept only emigration (or only immigration). As typical examples, it is found that due to the economic situation, a population can leave one region to inhibit some other regions or we can consider the fish leaving the rivers, moving to the sea while the sea fish remain locked in the sea .
These types of phenomena can be modeled with a reducible migration matrix, which we will discuss in this chapter. In addition, in this chapter we also consider complete birth and death matrices instead of diagonal structures. Since the appropriate Perron vector of Cre represents the stable patch structure, we approximate the original system exactly as we did in Chapter 5, that is:
Along with these general mortality and birth matrices, our next attempt is to weaken the irreducibility assumption on the migration matrix and to analyze the behavior with reducibleC. For the rest of the work, we will again use the same notation C to denote reducible migration matrix.
The Reducible Migration Matrix
Spectral Projections
In this section, we will prepare the ground for defining spectral projection operators that will be applied to the perturbed system for asymptotic analysis. We define a basis set for the null space of the migration operator Can also construct the basis set for the adjacent null space of C. In the rest of the work, we will assume that our migration matrix C depends on age, but that its normal form remains unchanged as it varies.
If our block has only nonzero diagonal elements, then we have the following explicit expressions of the corresponding left base. GENERALIZATIONS OF THE DISTURBED MODEL 60 since C is a Kolmogorov matrix, the sum of the elements of a certain column is equal to zero.
Model with General M, B and Reducible C
Hypotheses & Properties
To prove the second statement, we note thatej enj, . . ,0) for 1≤ j ≤m, in which the Perron vectors corresponding to the irreducible matrices Cnj and thus differentiable-.
Lifting Theorem
The first statement is obvious since the determinant of CS(a) is twice differentiable and bounded from zero by uniform invertibility of CS(a). Thus the assumptions of the Trotter formula, [20, Corollary III 5.8], are satisfied and therefore the type (G(t))t≥0 is the same as that of the semigroup generated by (S,D(S)) . GENERALIZATIONS OF THE DISTRIBUTED MODEL 62 to handle inhomogeneous boundary conditions u(0,t) = Bu+g where is a vector, possibly depending on time.
There are several versions of tracking theorems that can remove the inhomogeneity from the boundary to the interior, but here the problem is complicated due to the presence of a small parameter. Since S is the diagonal differentiation with respect to a, (6.9) is just the Cauchy problem for the system of linear non-autonomous equations ua = Q(a)u, where Q(a) :=−λI−M(a )+ 1C(a). Since the construction above does not depend on , you have the same regularity intasfmet bounds on derivatives independent of .
Since u is the solution of the Cauchy problem for the differential equation ina, it is differentiable with respect to a. We note that in this approach, upwelling produces its own time derivative on the right-hand side of the equation, causing some difficulty in the asymptotic analysis.
Formal Asymptotic Expansion
Bulk Approximation
In the boundary part, the comparison of order terms in the hydrodynamic subspace (ie, from equation (6.23)) we have. To write the system (6.30) as an abstract Cauchy problem, we introduce the operator Aas. Here we also note that. which implies that the sum of the hydrodynamic space scalars gives the total population.
From the above set of error equations, we observe that in the initial and boundary equations have some expressions which are not in order. To remedy the situation, we need to introduce corrections that will take care of the transient phenomena occurring close to t = 0 and to the limit a = 0. They should not 'break' the approximation away from spatial and temporal boundaries and should therefore rapidly decrease to zero with increasing distance from both boundaries.
Initial Layer
GENERALIZATIONS OF THE DISTRIBUTED MODEL 74 where we defined ˜w0(a,0) := w0(a) to get rid of w0(a) from the initial error equation of the bulk kinetic part (6.33). So inS all the eigenvalues of CS have negative real parts and therefore decay ˜w0 to exponentially fast as expected from a low term. We will also note that there are additional errors on the boundary due to initial layer.
GENERALIZATIONS OF THE DISTRIBUTED MODEL 75 We can see that O(1) terms remain in boundary error equation as expected and also a new error term introduced in boundary due to initial layer.
Boundary Layer
Thus to eliminate the largest term at the boundary, the boundary layer must be the solution. We note that, even with the boundary layer, we still have terms depending on / and this requires the introduction of the corner layer.
Corner Layer
Considering the problem mentioned in chapter 6, it turns out that we have to work with easy solutions of equations. First, from Lemma 6.8 we know that for sufficiently large λ there exists a classical solution of the stationary problem. In this case the basic solution matrix of the equation z0a(a) = CS(0)z(a) is simply the exponential of the matrix:.
In the following, the constants depend only on the coefficients of the problem and T, but not on the initial data. INTEGRAL FORMULATION 95 are bounded by c8(T)kφmkX, where c8(T) is related to the solution type ¯unj. Using the projections Pj, where 1 ≤ j ≤ 2, and Q, we can derive the corresponding expressions of the boundary equations on the hydrodynamic space as
Following [28] and using the Hille-Yosida theorem, we proved that the perturbed problem was well-posed. CONCLUSION 109 this assumption is the existence of a unique (up to scalar multiple) positive eigenvector corresponding to the dominant eigenvalue of the migration matrix. The results of the former correspond to our final error estimates described in Theorem 7.3.
In our case, the absence of the boundary and corner layers in the final approximation is due to the choice of the state space L1(R+,Rn).
