Applications of Perron-Frobenius theory to population dynamics

Li, Chi-Kwong; Schneider, Hans

doi:10.1007/s002850100132

Cited by 133 publications

(94 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, we restrict ourselves to constant matrices, i.e., focus on projection rather than prediction (Caswell, 2001, p. 30). The basic reproduction ratio is now defined as (Cushing and Yicang, 1994;Caswell, 2001;Diekmann et al, 1990;Li and Schneider, 2002). Here ρ denotes the spectral radius (the largest modulus of all eigenvalues), I the identity matrix and the exponent −1 matrix inversion.…”

Section: Model and Resultsmentioning

confidence: 99%

“…Here ρ denotes the spectral radius (the largest modulus of all eigenvalues), I the identity matrix and the exponent −1 matrix inversion. The matrix G = [g lk ] = F(I−S) −1 is sometimes called fundamental matrix (Caswell, 2001) or next-generation matrix (Diekmann et al, 1990;Li and Schneider, 2002). Each entry g lk gives the expected number of offspring in birth state l that are born over its life time to an individual that was itself born in birth state k. Thus, the matrix G has as many rows with non-zero entries as there are birth states.…”

Section: Model and Resultsmentioning

confidence: 99%

“…An alternative method to evaluate population growth is to compute the basic reproduction ratio R 0 (Diekmann et al, 1990;Cushing and Yicang, 1994;Caswell, 2001;Li and Schneider, 2002). R 0 can be computed as the dominant eigenvalue of the next-generation matrix G = [g lk ], where the matrix entries g lk equal the expected number of offspring in birth state l born over its life time to an individual with birth state k. Importantly, R 0 has the property…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Necessary and sufficient conditions for $$R_{0}$$ to be a sum of contributions of fertility loops

Rueffler

Metz

2012

J. Math. Biol.

View full text Add to dashboard Cite

As a present to Odo Diekmann for his 65th birthday: the answer to a question that he posed in a conversation sometime ago.Abstract Recently, de-Camino-Beck and Lewis (2007, Bull Math Biol 69:1341-1354 have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R 0 in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graphreduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases R 0 allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single in- Rueffler and Metz dividual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R 0 that may sometimes be used to good effect in cases where no simple formula for R 0 exists.

show abstract

Section: Model and Resultsmentioning

confidence: 99%

Section: Model and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Necessary and sufficient conditions for $$R_{0}$$ to be a sum of contributions of fertility loops

Rueffler

Metz

2012

J. Math. Biol.

View full text Add to dashboard Cite

show abstract

“…Susceptible humans in patch k spend a proportion P ki of their time in patch i. Once the human resident of patch k is infected by the mosquito, it can in turn infect a susceptible mosquito that is visiting patch k, but resides in patch j. Mosquito residents of patch j, spend a proportion Q jk of their time in patch k. Now summing over all possible patches k, shows that the original infected mosquito in patch i can cause a secondary mosquito infection in patch j, provided that the expression in (22), which represents the (j, i)th entry of the product QP , is positive, as claimed. Irreducibility of the matrix QP therefore means that a single infected mosquito resident in any patch, has the potential to cause a mosquito infection in any other patch later on, although the latter infection is no longer necessarily a secondary infection, but may occur through a finite number of consecutive mosquito-human-mosquito infections as described above.…”

Section: First What Does Irreducibility Of P Q and Qp Mean?mentioning

confidence: 99%

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

Ruktanonchai

Smith

Leenheer

2016

Mathematical Biosciences

View full text Add to dashboard Cite

We consider the dynamics of a mosquito-transmitted pathogen in a multipatch Ross-Macdonald malaria model with mobile human hosts, mobile vectors, and a heterogeneous environment. We show the existence of a globally stable steady state, and a threshold that determines whether a pathogen is either absent from all patches, or endemic and present at some level in all patches. Each patch is characterized by a local basic reproduction number, whose value predicts whether the disease is cleared or not when the patch is isolated: patches are known as "demographic sinks" if they have a local basic reproduction number less than one, and hence would clear the disease if isolated; patches with a basic reproduction number above one would sustain endemic infection in isolation, and become "demographic sources" of parasites when connected to other patches. Sources are also considered focal areas of transmission for the larger landscape, as they export excess parasites to other areas and can sustain parasite populations. We show how to determine the various basic reproduction numbers from steady state estimates in the patched network and knowledge of additional model parameters, hereby identifying parasite sources in the process. This is useful in the context of control of the infection on natural landscapes, because a commonly suggested strategy is to target focal areas, in order to make their corresponding basic reproduction numbers less than one, effectively turning them into sinks. We show that this is indeed a successful control strategy -albeit a conservative and possibly expensive one-in case either the human host, or the vector does not move. However, we also show that when both humans and vectors move, this strategy may fail, depending on the specific movement patterns exhibited by hosts and vectors.

show abstract

“…It also appears in life history theory (Metz et al 1992;Mylius and Diekmann 1995), age-or stage-structured population dynamics (Caswell 2001), and invasion biology (de Camino-Beck and Lewis 2007). In all cases, if R 0 >1 a small population will establish/persist but if R 0 <1 then individuals cannot on average replace themselves and the population declines to extinction (Cushing and Zhou 1994;Li and Schneider 2002). Note that R 0 theory is different and more general than R* theory, which predicts competitive exclusion based on minimal resource requirements (Tilman 1982), but which breaks down under dispersal (Abrams and Wilson 2004) and is limited to models of resource competition.…”

Section: Introductionmentioning

confidence: 99%

An R 0 theory for source–sink dynamics with application to Dreissena competition

Krkošek

Lewis

2009

Theor Ecol

View full text Add to dashboard Cite

Source-sink dynamics may be ubiquitous in ecology. We developed a theory for source-sink dynamics using spatial extensions of the net reproductive value, R 0 , which has been used elsewhere to define fitness, disease eradication, population growth, and invasion risk. R 0 decomposes into biologically meaningful componentslifetime reproductive output, survival, and dispersal-that are widely adaptable and easily interpreted. The theory provides a general quantitative means for relating fundamental niche, biotic interactions, dispersal, and species distributions. We applied the methods to Dreissena and found a resolution to a paradox in invasion biology-competitive coexistence between quagga (Dreissena bugensis) and zebra (D. polymorpha) mussels among lakes despite extensive niche overlap within lakes. Source-sink dynamics within lakes between deepwater and shallow habitats, which favor quagga and zebra mussels, respectively, yield a metacommunity distribution where quagga mussels dominate large lakes and zebra mussels dominate small lakes. The sourcesink framework may also be useful in spatial competition theory, habitat conservation, marine protected areas, and ecological responses to climate change.

show abstract

Applications of Perron-Frobenius theory to population dynamics

Cited by 133 publications

References 19 publications

Necessary and sufficient conditions for $$R_{0}$$ to be a sum of contributions of fertility loops

Necessary and sufficient conditions for $$R_{0}$$ to be a sum of contributions of fertility loops

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

An R 0 theory for source–sink dynamics with application to Dreissena competition

Contact Info

Product

Resources

About