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Competing Size-Structured Species
Published in Ovide Arino, David E. Axelrod, Marek Kimmel, Mathematical population dynamics, 2020
This general model includes the famous matrix model for age-structured populations of Leslie (1945). In this case the classes are age classes one time unit in length, so that fj = 1, fij = 0 if j ≠ i − 1, and fi,j−1 = 1, (i.e., all surviving individuals must advance one age class in one unit of time). Ρ then becomes a Leslie matrix. Another special case occurs when 0 < fj = constant, fij = 0 if j ≠ i − 1, and fi,j−1 = 1 (i.e., individuals either remain in their class or move to the next class in one unit of time). In this case Ρ is an Usher matrix. Usher matrix models have found extensive use in size-structured models of tree forest dynamics (see, e.g., Ek and Monserud, 1979; Usher, 1972).
Characterization of Systems
Published in Ferenc Szidarovszky, A. Terry Bahill, Linear Systems Theory, 2018
Ferenc Szidarovszky, A. Terry Bahill
This model was originally introduced by Leslie [27], and matrix A in (3.115) is usually called the Leslie-matrix. The population structure can be predicted for any future time by solving the governing difference equation. Some properties of this system will be analyzed in later chapters.
Stage-structured control on a class of predator-prey system in almost periodic environment
Published in International Journal of Control, 2020
Tianwei Zhang, Li Yang, Lijun Xu
In practice, animal populations are often measured by size with age structure used as an approximation to size structure. The study of age-structured models is considerably simpler than the study of general size-structured models, primarily because age increases linearly with the passage of time while the linkage of size with time may be less predictable. Age-structured models may be either discrete or continuous. For example, in 1926, McKendrick (1926) obtained a linear continuous model, which first introduced age structure into the dynamics of a one-sex population. Leslie (1945) considered a model for the growth of the number of females in an animal population (e.g. a population of rats), and formulated a linear discrete model, which is usually known as the Leslie matrix model. Similar forms appeared in earlier work of Bernardelli (1941) and Lewis (1942). In order to describe the behaviour of individuals with different ages, Landahl and Hanson (1975) and Tognetti (1975) proposed the named stage structure models by using different equations. In the real world, almost all animals have the stage structure of immature and mature, and species at two stages may have different behaviours. For many organisms, immature and mature individuals are ecologically very different and thus occupy different niches, different positions in their local food web, and may even occupy distinct habitats and foodwebs (e.g. terrestrial amphibians with aquatic larvae (Conrad, 1972; Leff & Bachmann, 1988); butterflies and caterpillars (Brauer & Castillo-Chavez, 2012; Stamp & Bowers, 1990); mosquitoes that are vectors for malaria (Brauer & Castillo-Chavez, 2012; Kenneth, 1948); adult trees and seedlings (Ma, 1996; Neuner, Bannister, & Larcher, 1997)). In recent years, the stage-structured models have received much more attention, see Kuang and So (1995), Zhang and Gan (2013), Zhang, Li, and Ye (2011), Xu, Chaplain, and Davidson (2005), Chen and You (2008), Kawachi (2008), Liu, Zhang, and Zhou (2014), Sacker (2011), and the references therein. In Zhang et al. (2011) Cui and Song proposed the following periodic predator-prey model with stage structure for prey: where and denote the density of immature and mature prey species, respectively, and y is the density of the predator that preys on . Under the assumption that the coefficients of system (1) are all continuous positive ω-periodic functions, the authors obtained a set of sufficient and necessary conditions which guarantee the permanence of system (1).