In earlier times in the US there was a kids rhyme that went '... way down yonder in the pawpaw patch' and maybe it's still sung in some rural areas where there are native stands of Asimina triloba that haven't succumbed to urban sprawl or agricultural clearing. It's called pawpaw there and not to be confused with what Australians usually call pawpaw, namely Carica papaya. The species is the largest edible tree fruit native to North America and although it has commercial potential it still has not really taken off there or elsewhere, even though the fruit is pleasant tasting; Brix is 15-20% when ripe. The USDA maintains a germplasm collection at Kentucky State University for breeding work and other studies with the goal that it might be developed into a more important crop in the future. The 'pawpaw patch' ditty stems from the fact that plants sucker freely in suitable climates and conditions so that whole clumps of trees that were commonly seen in the past were in fact clones of a single or few trees. Compared to more commercially-successful fruit tree species there has been far less research on its properties and behaviour, and as a result there are various confusing statements regarding reproduction processes - flowering properties, self-compatibility, the need for cross-pollination, whether fruit yield is pollinator and/or resource limited, seed viability, possible yields etc.
If we consider first the question of genetic self- and cross-compatibility. Flowers in Asimina are hermaphrodite and exhibit protogynous dichogamy, a remarkably common feature of pollination biology in the Annonaceae family. Inflorescences occur as single axillary flowers with each gynoecium having 3-9 ovaries and each of these containing 5-20 ovules; fertilisation of these ovaries can lead to multi-seeded fruit being formed in clusters. Most dichogamous fruiting species have a two day cycle, with in the case of protogynous behaviour the female organs being effective on one day and male the following with an interim period where neither sexual phase is active. However for this species the process is greatly slowed down, with stigmas being receptive for 3-15 days and anther release of viable pollen occurring for 2-3 days afterwards. Such slower dichogamous cycling is more common in temperate species like Asimina than in tropical and sub-tropical species. Temporal separation favours cross-pollination and can minimise inbreeding depression and the possible build-up of deleterious mutations and alleles in progeny. However depending on environment, there is sometimes a brief overlap period of 1-2 days where both organs are effective, and self- pollination then becomes possible.
Flower production in the species is prolific, but natural pollination (usually by beetles with smaller contributions from flies and thrips) in single trees results in low fruit set and yield (2-4%) and there is little improvement when done by hand. Fruit abortion following fruit set is not usually a problem, but seed counts in these fruit are low (3-5), with many aborting or being sterile. This low productivity is not the result of inadequate pollen supply as hand pollination can deliver 30-70 pollen grains per stigma. Pollen is released from anthers as tetrads and germination requires a supportive stigma environment and secretions, with normally only one of these four grains forming a pollen tube that grows towards an ovary. Many of these abort along the way and do not result in fertilisation. Together these findings indicate there is a degree of genetic self-incompatibility in the species, probably polygenic, and unlike most species in the Annonaceae family. There are a few named cultivars where self-fertilisation appears to be less of a problem, but yield, fruit quality and seed count is still less than what can be achieved with cross pollination. Naturally-mediated cross-pollination can produce 5-10% fruit yield, and this can be increased to 10-40% with hand pollination, indicating yield is normally pollination limited. A separate factor limiting yield is maternal resource availability through tree health and vigour, as larger stems support more fruit through to maturity.
Currently (2020) there are 11 recognised species in the genus Asimina, with several natural hybrids between them. Molecular genetic work has established the genus can be divided into three taxonomic clades, with A parvifolia (the Dwarf Pawpaw) and A triloba both in the same clade. This close relationship is supported by many common morphological features between the two, and a natural hybrid in North America has been characterised and officially recognised (A X piedmontana). Parallel research on parvifolia reproduction (out-crossing, pollination limitation, self-incompatibility etc) has shown similar behaviour to triloba.
Asimina is therefore a facultative xenogamous species strongly favouring cross-pollination, with self-pollination essentially being a back stop that can produce a small number of fruit but with only a few seeds that are usually infertile. If you are intending to grow the species in the south west of WA you'll have more success if you plant (or graft) more than one plant from different genetic stock, and you should also be prepared to hand-pollinate to optimise quality and quantity of fruit. Named varieties may give better results but will still do best with cross-pollination, and they're difficult to source in WA.