title: Approximation of invariant measures for a class of maps with indifferent fixed points

authors: Murray, Rua

subjects: mathematics

abstract: Certain dynamical systems on the interval with neutrally stable repelling points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, since they exhibit polynomial rather than exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper presents a class of these maps which are expanding (with convex branches) for which the invariant probability measures can be rigorously approximated by Ulam’s method (a sequence of finite rank approximations to the transfer operator). L1–convergence of the scheme is proved, and some numerical experiments are reported.

text: Approximation of invariant measures for a class of maps with indifferent fixed points Certain dynamical systems on the interval with neutrally stable repelling points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, since they exhibit polynomial rather than exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper presents a class of these maps which are expanding (with convex branches) for which the invariant probability measures can be rigorously approximated by Ulam’s method (a sequence of finite rank approximations to the transfer operator). L1–convergence of the scheme is proved, and some numerical experiments are reported.

year: 2005

title: Thresholds for sustainable regeneration in urban restoration plantings in Hamilton City, New Zealand

authors: Overdyck, Elizabeth

subjects: forest restoration

abstract: Urban forest patches have unique environmental and landscape characteristics which may influence the restoration of native plant communities. Urbanisation can lead to a drier and warmer climate, a prevalence of exotic seed sources and isolation from remnant native forest seed sources. This research investigates how these factors influence native species presence in different aged urban forest patches and uses life history traits to identify vulnerable species groups which may require active reintroduction. Seed rain, soil seed banks and vegetation composition was recorded within urban forest restoration plantings (10-36 years old) in Hamilton City, New Zealand with comparison to naturally regenerating forest within the city and a nearby rural forest remnant. To address dispersal limitation for several key mid to late successional forest species an experiment was also undertaken to investigate broadcast seeding as a method to reintroduce trees with large seeds and fleshy fruits into established early successional vegetation.

Seed rain, soil seed banks (fern spores inclusive) and understorey vegetation in urban forest were found to have higher exotic species richness and lower native species density and richness than rural forest. Both understorey vegetation and soil seed banks of urban sites >20 years old had lower exotic species richness than younger (10–20 years) sites, indicating a developmental threshold that provided some resistance to exotic species establishment. A prevalence of exotic species in urban seed rain, however, will allow reinvasion through edge habitat and following any disturbance to canopy vegetation. Persistent soil seed banks from both urban and rural sites were dominated by exotic herbaceous species and native fern species, while few other native forest species were found to persist for >1 year in the seed bank.

Urban native seed rain was greater in quantity than exotic seed rain (reflecting immediately surrounding vegetation) although only when native canopy species had been planted suggesting a benefit of initial planting to encourage restoration of native communities. Novel species arriving in the seed rain, but not present in the immediate vegetation, were often not abundant in quantity but represented three quarters of the native species recorded in urban seed rain providing evidence for some long-distance dispersal (particularly for wind-dispersed species) and potential for new species to establish. Urban and rural seed rain contained a similar number of novel native species arriving, although compositionally dissimilar, whereas a greater number of novel exotic species arrived in urban seed rain. Establishment for some native species arriving in urban seed rain was limited, e.g. ferns, indicating a suitable forest microclimate is still to develop. 

It was found that the native forest flora in Hamilton City represented just over half (57%) of the species present in forests of the broader Hamilton Ecological District. This suggests limited natural colonisation from beyond the urban area and the absent species are suggested as priorities for urban reintroduction. In turn only 35% of the city forest flora was found to be represented in the seed supply (annual seed rain and soil seed bank) and understorey sampling in urban forest patches. An over representation of trees in the city forest flora may reflect some relictual long-lived species that are surviving but may no longer have viable populations. Forbs and parasitic plants, highly shade tolerant (i.e. late successional) species and those with biotic pollinators were under represented in the seed supply and understorey indicating some limitation for regeneration or colonisation in young urban forests. The richness of bird-dispersed native species in urban forest patches increased with proximity and size of good quality native vegetation but no other effects of dispersal mode on urban native species presence were found. 

To facilitate dispersal, broadcast seeding was found to be a viable method of improving regeneration of large-seeded late successional trees and may be a cost-effective alternative to planting saplings. Seedling establishment can be improved with fruit flesh removal and clay ball treatments, especially in the presence of mammalian seed predators.

text: Thresholds for sustainable regeneration in urban restoration plantings in Hamilton City, New Zealand Urban forest patches have unique environmental and landscape characteristics which may influence the restoration of native plant communities. Urbanisation can lead to a drier and warmer climate, a prevalence of exotic seed sources and isolation from remnant native forest seed sources. This research investigates how these factors influence native species presence in different aged urban forest patches and uses life history traits to identify vulnerable species groups which may require active reintroduction. Seed rain, soil seed banks and vegetation composition was recorded within urban forest restoration plantings (10-36 years old) in Hamilton City, New Zealand with comparison to naturally regenerating forest within the city and a nearby rural forest remnant. To address dispersal limitation for several key mid to late successional forest species an experiment was also undertaken to investigate broadcast seeding as a method to reintroduce trees with large seeds and fleshy fruits into established early successional vegetation.

Seed rain, soil seed banks (fern spores inclusive) and understorey vegetation in urban forest were found to have higher exotic species richness and lower native species density and richness than rural forest. Both understorey vegetation and soil seed banks of urban sites >20 years old had lower exotic species richness than younger (10–20 years) sites, indicating a developmental threshold that provided some resistance to exotic species establishment. A prevalence of exotic species in urban seed rain, however, will allow reinvasion through edge habitat and following any disturbance to canopy vegetation. Persistent soil seed banks from both urban and rural sites were dominated by exotic herbaceous species and native fern species, while few other native forest species were found to persist for >1 year in the seed bank.

Urban native seed rain was greater in quantity than exotic seed rain (reflecting immediately surrounding vegetation) although only when native canopy species had been planted suggesting a benefit of initial planting to encourage restoration of native communities. Novel species arriving in the seed rain, but not present in the immediate vegetation, were often not abundant in quantity but represented three quarters of the native species recorded in urban seed rain providing evidence for some long-distance dispersal (particularly for wind-dispersed species) and potential for new species to establish. Urban and rural seed rain contained a similar number of novel native species arriving, although compositionally dissimilar, whereas a greater number of novel exotic species arrived in urban seed rain. Establishment for some native species arriving in urban seed rain was limited, e.g. ferns, indicating a suitable forest microclimate is still to develop. 

It was found that the native forest flora in Hamilton City represented just over half (57%) of the species present in forests of the broader Hamilton Ecological District. This suggests limited natural colonisation from beyond the urban area and the absent species are suggested as priorities for urban reintroduction. In turn only 35% of the city forest flora was found to be represented in the seed supply (annual seed rain and soil seed bank) and understorey sampling in urban forest patches. An over representation of trees in the city forest flora may reflect some relictual long-lived species that are surviving but may no longer have viable populations. Forbs and parasitic plants, highly shade tolerant (i.e. late successional) species and those with biotic pollinators were under represented in the seed supply and understorey indicating some limitation for regeneration or colonisation in young urban forests. The richness of bird-dispersed native species in urban forest patches increased with proximity and size of good quality native vegetation but no other effects of dispersal mode on urban native species presence were found. 

To facilitate dispersal, broadcast seeding was found to be a viable method of improving regeneration of large-seeded late successional trees and may be a cost-effective alternative to planting saplings. Seedling establishment can be improved with fruit flesh removal and clay ball treatments, especially in the presence of mammalian seed predators.

year: 2014

title: Maori and Pakeha at Te Aroha: the context: 2: Maori in Hauraki in the nineteenth century

authors: Hart, Philip

abstract: After the arrival of Europeans, the Maori population of Hauraki suffered a rapid decline. Some rangatira opposed Pakeha ways, whereas others adopted these for their personal benefit. Keeping ‘the peace of Hauraki’ required government agents to intervene in various disputes between hapu (Ngati Hako in particular causing concern in the 1870s and early 1880s). Although rangatira had links to both Queen Victoria and King Tawhiao, the government was relieved that most remained ‘loyal’ to the new order, and only an intransigent minority opposed the spread of ‘civilisation’ through its roads, telegraph, and the snagging of the Waihou River. A liking for Pakeha goods encouraged collaboration, with Maori joining the cash economy through their involvement in road making, gum digging, and European agriculture to raise money for, in part, traditional gatherings that for a while were more lavish than earlier possible.
Maori of all ranks were quick to stand up for their rights by using the court system, reminding Pakeha of the Treaty of Waitangi, and, in some cases, violence. Tensions were eased by Maori socializing with Pakeha in sport, horse racing, and even the Volunteers, but drinking together in hotels could result in fights, and the lure of alcohol had to be countered by temperance movements. A few Maori children attended school, with Pakeha, and for a time the government provided a (free) doctor and vaccinated them against smallpox (though some preferred traditional ‘cures’ for other ailments). For most, living conditions remained poor. Criminal behaviour was of a minor nature. Christianity had to compete with old beliefs (notably in maketu), newer ones such as ‘Hauhauism’, and by the later years of the century the popularity of the Mormons.
Examples are included of intermarriage, ‘half-castes’, and Pakeha Maori, all being notable features of the time. In a variety of ways, Maori society was sufficiently resilient to adapt and thereby to survive the impact of Pakeha settlement, which produced massive changes and dominated the region well before the end of the century.

text: Maori and Pakeha at Te Aroha: the context: 2: Maori in Hauraki in the nineteenth century After the arrival of Europeans, the Maori population of Hauraki suffered a rapid decline. Some rangatira opposed Pakeha ways, whereas others adopted these for their personal benefit. Keeping ‘the peace of Hauraki’ required government agents to intervene in various disputes between hapu (Ngati Hako in particular causing concern in the 1870s and early 1880s). Although rangatira had links to both Queen Victoria and King Tawhiao, the government was relieved that most remained ‘loyal’ to the new order, and only an intransigent minority opposed the spread of ‘civilisation’ through its roads, telegraph, and the snagging of the Waihou River. A liking for Pakeha goods encouraged collaboration, with Maori joining the cash economy through their involvement in road making, gum digging, and European agriculture to raise money for, in part, traditional gatherings that for a while were more lavish than earlier possible.
Maori of all ranks were quick to stand up for their rights by using the court system, reminding Pakeha of the Treaty of Waitangi, and, in some cases, violence. Tensions were eased by Maori socializing with Pakeha in sport, horse racing, and even the Volunteers, but drinking together in hotels could result in fights, and the lure of alcohol had to be countered by temperance movements. A few Maori children attended school, with Pakeha, and for a time the government provided a (free) doctor and vaccinated them against smallpox (though some preferred traditional ‘cures’ for other ailments). For most, living conditions remained poor. Criminal behaviour was of a minor nature. Christianity had to compete with old beliefs (notably in maketu), newer ones such as ‘Hauhauism’, and by the later years of the century the popularity of the Mormons.
Examples are included of intermarriage, ‘half-castes’, and Pakeha Maori, all being notable features of the time. In a variety of ways, Maori society was sufficiently resilient to adapt and thereby to survive the impact of Pakeha settlement, which produced massive changes and dominated the region well before the end of the century.

year: 2016