Back to Search
Start Over
Spectral properties of a non-compact operator in ecology
- Source :
- Journal of mathematical biology. 82(6)
- Publication Year :
- 2020
-
Abstract
- Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate $$\lambda $$ , or a stable-stage distribution $$\psi $$ . In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector $$\psi $$ corresponding to its spectral radius $$\lambda $$ , this $$\lambda $$ is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population $$\varphi _0$$ , there is a $$c>0$$ such that $$T^n\varphi _0/\lambda ^n \rightarrow c \cdot \psi $$ .
- Subjects :
- Perron–Frobenius theorem
0303 health sciences
education.field_of_study
Pure mathematics
Distribution (number theory)
Ecology
Spectral radius
Applied Mathematics
Essential spectrum
Population
Compact operator
01 natural sciences
Agricultural and Biological Sciences (miscellaneous)
Models, Biological
010305 fluids & plasmas
03 medical and health sciences
Operator (computer programming)
Modeling and Simulation
Bounded function
0103 physical sciences
Animals
education
030304 developmental biology
Mathematics
Subjects
Details
- ISSN :
- 14321416
- Volume :
- 82
- Issue :
- 6
- Database :
- OpenAIRE
- Journal :
- Journal of mathematical biology
- Accession number :
- edsair.doi.dedup.....158ff8f498f1c8e21fc4bc9e668aeb94